THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE ONSET OF LAMINAR NATURAL CONVECTION IN A FLUID WITH HOMOGENEOUSLY DISTRIBUTED HEAT SOURCES Walter Ralph Debler A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1959 June, 1959 IP-374
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 Doctoral Committee: Professor John A. Clark, Co-chairman Professor Chia-Shun Yih, Co-chairman Professor Robert C. F. Bartels Professor Frankl LO Schwartz Professor Gordon J. Van Wylen ii
 AC KNOWLEDGEMENTS The author wishes to express his gratitude to Professors John A. Clark and Chia-Shun Yih who, along with the other members of his doctoral committee, generously gave suggestions and encouragement during the course of the work presented herein. A debt of gratitude is owed to Dean John S. McNown, University of Kansas, who as a professor at the University of Michigan shared with the students his wide knowledge of fluid mechanics and stimulated the author to study further in the subject. The help and resourcefulness of the technicians from the Department of Mechanical Engineering and the Department of Engineering Mechanics during the construction of the experimental apparatus is much appreciated. The author was assisted financially in his work by a Rackham Fellatiship and a research assistantship for a project sponsored by the Office of Ordnance Research under Contract No. DA-20-018-ORD-15556 with the University of Michigan. The two copper plates used in the experimental work were kindly donated by the American Brass Company of Detroit, Michigan.
 TABLE OF CONTENTS Page ACKNOWLD GEMENTS iii LIST OF TABLES -vi LIST OF FIGURES vii LIST OF APPENDICES ix CHAPTER I INTRODUCTION 1 1. Statement of the Problem I 2, Historical Background 2 3, Related Literature 4 CHAPTER II MATHEMATICAL SOLUTION OF THE PROBLEM 6 1. The Governing Equations 6 2. Method of Solution 20 3. Temperature and Velocity Distribu~ tion in the Fluid Layer 29 4. Heat Transfer Aspects of the Solut on 40 5. Review of Assumptions 45 CHAPTER III MATHEMATICAL FORMULATION AND SOLUTION OF RELATED STAB ILITY PROBLEMS 52 1. Rigid Thermally Conducting Upper Boundary and Rigid Thermally Insulating Lower Boundary 52 2, Free Thermally Conducting Upper Surface and Rigid Thermally Insulating Lower Boundary 61 3o Other Related Problems 65 4o Problems Similar to Those Discussed 66 CHAPTER IV EXPBRIMENTAL VERIFICATION OF ANALYSIS 69 1. Description of Apparatus 69 2, Exper imental Procedure 73 3. Results of Exper iment 74 iv
 TABLE OF CONTENTS (CONTtD) CHAPTER V RESULTS OF THEORECTICAL ANALYSIS AND EXPERIMENTAL INVEST IGAT ION 77 CHAPTER V I CONCLUS IONS 81 V~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 LIST OF TABLES Table Page I Summary of Computations of Neutral- 28 Stability Curve for a Fluid Having Homogeneously Distributed Heat Sources Contained between Hortizontal Boundar ies that are Rigid and Isothermalo,.-*...* II Summary of Experimental Data -... 75 vi
 LIST OF F IGIRES 'jgte P' Schematic Drawing of Fluid Layer with the Mean Temperature Distri"but ion and the Coordinate System (Problem II) 7 2 Neutr alcstab il ity Curve for Laminar Natural Convection in a Pluid with Homogeneously 1 istr ibuted Heat Sources Contained between Ho izontal Boundaries that are R1igid and Isothermal (Problem II) 27 3 The Varat ion of the Non-dimens ional Temperature and Velocity Functions, 'T and w ~ with the NonzDimensiona1 Vertical Coordinate9 (Problem II) 30 4 - The Cell Pattern in one Plane of Symmetry at the Onset of Convection for the First Mode and for an Assumed Hexagonal Tesselationo (Problem II) 41 5 The Ratio of the Laplacian of the Vertical Velocity Component to 'the Der ivative, in the Vertical Direction9 of the Divergence of the Velocity0 51 6 Schematic Drawing of Fluid Layer with Mean Temperature Distribution and Coord inate System, (Problem III 1) 52 7 Neutral-stab ility Curve for Laminar Natural Convection in a Fluid with Homogeneously Distributed Heat Sources Contained between a Lower Adiabat ic Rigid Boundary and an Upper Rigid Isothermal Boundary. (Problem IIIol) 59 8 The Varation of the Non-dimensional Temperature and Vel.ocity Punctions TA and ~V with the Nonddiomensional Vertical Coordinate9, o (Problem IITol) 60 9 Schematic Drawing of Fluid Layer with the Mean Temperature Distribution and the Coordinate System, (Problem IIIo2) 61 10 Neutralstability Curve for Laminar Natural Convection in a Fluid with Homogeneously Distributed Heat Sources Contained between a Lower Adiabatic Rigid Boundary and an Upper Free isothermal Boundary0 (Problem III.2) 64 voi
 LIST OF F IGURES 11 Temperature-dens ity Relationship for Water near 40 CO 67 12 Schematic Diagram of Experimental Apparatus 70
 APPENDICES Appendix Page I Evaluation of Integrals with Trigonometric and Hyperbolic Integrands. 83 II Simplification of Secular Determinant for Solution of a Particular Instability Problem. 87 III Coefficients for Approximative Solution to Problems Concerning the Instability of a Fluid with Homogeneously Distributed Heat Sources. 90 IV Solution for the Thermal Instability of Water at 40 Celsius. 92 ix
 CHAPTER I - INTRODUCTION 1. Statement of the Problemo The study of the thermal instability (onset of natural convection) of a fluid having homogeneously distributed heat sources is of interest as it has some bearing to the design of nuclear reactorso Of equal importance is the fact' that when results of such a study are compared with other investigations, a conclusion as to the qualitative influence of the variation of the temperature gradient on. the general problem can be obtainedo In the present work the stability of a viscous fluid confined between two horizontal boundaries and with homogenieously distributed heat sources is examined0 The fluid under consideration has the state of rest as an equilibrium state, but can become unstable if a critical val"ue 'of the Rayleigh number is exceeded0 The ensuing motion is one of maintained natural convection~ The Rayleigh number is defined to be go b3(A~)/ p, in which g is the gravitational constant, oC the coefficient of thermal expansion9 b the depth of the fluid layer, A~ the temperature difference between the center of the fluid and one of its boundaries, p the kinematic viscosity9 and k the thermal diffusivity. The value of the critical Rayleigh number is shown to depend upon the nate of the of thfluids boundaries (e.g., free or rigid; conducting or adiabatic)0.~~~~~~~~~~~~~~~~~~~~~~~t..
 A mathematical solution of the problem has been achieved by employing an approximation technique9 and an experiment has been performed for one boundary configuration to verify the analysiso 2~ Histor ical Background. Stability problems in the field of fluid mechanics have occupied the attention of a host of authors since the latter part of the nineteenth century, The scope of the investigations covers the many aspects of' secondary laminar flow and init ial transition to turbulence. The influence of velocity profile9 boundary shape, acceleration, pressure gradient, temperature variation and magnetic fields, are but some of the facets of the general problem which have received cons ider at ion. Spec if ic appl icat ions of the obtained results are employed by aeronautical engineers and naval architects. The improvement of lift and the lessening of drag through boundary layer control has been the result of extensive experimental and mathematical work dealing with the stability of the flow in the boundary layer. Our knowledge of convective heat transfer has been broadened by including the influence of temperature variation on the fluid motion and studying the resulting stability problem, In this case the temperature variations cause a density variation which in turn can bring about natural convection, Meterologists are combining the thermal and rotational effects on secondary
 flow to gain an insight into the mechanisms of storm systems and the jet stream, Scientists interested in the earthts magnetic field are studying the secondary flows that could be induced in the earth's molten core, These geophysicists, along with astr ophysicists studying the atmosphere around the planets and physicists engaged in developing nuclear reactors utilizing an ionized plasma have done much to develop the science of magnetohydr odynamics The present thesis deals with a problem which is akin to that studied by Benard, Jeffreys, Low, Rayleigh and others0 The subject of that problem is the maintained convective motion in a fluid that is heated from below (hereafter referred to as the Benard problem)0 Pellew and Southwell (1) were able to obtain an exact solution for the mathematical formulation of the problem, while Chandra (2) and Schmidt and Mil3verton (3) previously obtained experimental data which are in agreement with the analytic solution. The equations for the Benard problem are directly related to the classical problem for the stability of viscous flow between rotating cylinders, first studied by Go I. Taylor (4), for the case in which the difference in radii of the two cylinders is small compared totheir mean, and the speeds of rotation of the cylinders are nearly equal0 Chandrasekhar (5) solved the resultant equations for these problems by an approximation technique even though
 the exact solution of Pellew and Southwell was already available - to show the power of the method and the rapid convergence of the attendant solution0 In a recent paper by Reid and Harris (6) the various exact and approximate methods of solution for the Benard problem are discussed with a view to giving some insight into the relative merits of the approximative methods that are available for solving those problems which do not have an exact solutiono For the problem that is presented in this thesis the method of Chandrasekhar (5) is utilized for solving the governing differential equations0 These equations do not lend themselves to an exact solution of the type given by Pellew and Southwello A study of the BBnard problem was of value in the work of this thesis not only because the method of solution 1s relevant but also because the quantitative results of that solution can be used to reveal the role played by the temperature distribution upon the stability of the fluid. For th'e case of the fluid being heated from below the mean temperature var iation is linear whereas for distributed heat sources the mean temperature var iation is par abolic if the thermal diffusivity is considered to be constanto 3, Related Literat.ure0,It was stated at the beginning of the previous section 09~bt~a
 that problems of hydrodynamic stability have attracted the interest of investigators in many fields of the physical sciences. To present a list of all the work that has been done in this area is not only impractical but also unnecessary, The book by Lin (7) is an excellent introduction to the subject and contains an extensive bibliography of 258 entries. A paper by Ostrach (8) deals with convection phenomena and makes reference to 38 other papers. A short summary of the work in heat transfer, including natural convection, that has been done in the last two years is presented by Eckert, Hartnett, and Irvine (9), The subject of magnetohydrodynamics has recently been outlined in book form by Cowling (10). In this field there have been new papers by Nisbet (11) and Yih (12). The work of Malkus (13) and Stuart (14) is significant for they obtain solutions for the equations governing hydrodynamic stability in which the non-linear terms have been retainedo With this work available, it is now possible (as suggested by Pellew and Southwell) to predict the shape of the convection cell, a fact that cannot be ascertained if the equations are linearized, * and which, therefore, must be arbitrarily assumed.
 CHAPTER I I MATHEMATICAL SOLUTION OF THE PROBLEM 1. The Governing Equations Consider a horizontal layer of fluid with a depth b, confined between two parallel planes x3 = O and x3 = b. These planes are solid, heat conducting and of equal temrperature. The fluid has uniformly distributed heat sources. The equations of motion and heat conduction applicable to this problem areand!D~ I-A~ Dt &mdash (2) in which the subscripts in the Navier-Stokes equations have values of 1, 2, and 3 for the coordinate directions, and the repetition of a particular suffix implies the summation convention of tensor analysis; Ut is the component of the fluid velocity in the L+h coordinate direction; p is the mass density of the fluid; X, a Lame constant; - the gravitational constant;- the pressure in the fluid; 7 and J), the dynamic and
 kinematic viscosities; [i the Kronecker delta, being zero for i 8 3 and unity for i = 3; and A the symbol for the Laplacian operator. Also, in the foregoing equations ~ is the absolute temperature of the fluid, 6 is the thermal diffusivity; - is the time rate of heat generation per unit volume of fluid, ~ is the thermal conductivity, and D is the substantial derivative: ot Dt 2 L t i at UXXx + U3 The coordinate system employed for this problem is presented in Fig. 1 along with the mean temperature distribution in the fluid layer. /X Heat Conducting, Solid Boundary X3. M b X Heat Conducting, Solid Boundary Fig. 1. Schematic Drawing of Fluid Layer with the Mean Temperature Distribution and the Coordinate System In writing equation (2) the dissipation of mechanical energy into heat is considered small and therefore neglected. This assumption will be reviewed later. The density of the fluid can be expressed as / /o { 4 - ~ (~- ~0)} (3) -7 -
 if the temperature differences in the fluid are small, In equation (3) p0 is the density at the reference temperature, ~0; and o( is the coefficient of thermal expans ion0 o Under equilibrium conditions the Ifluid layer is quiescent and the heat liberated by the fluid is carried to the boundarieLs by conduction alone0 Accordingly, equations (1) and (2) can be solved for this condition and by assuming constant thermal properties one has U[li - 0 (4):=2 ibX3- 3) + o (5) _m 1tK ~ (3(6) (x~i r~- "~'l- (7) in which the bar over a symbol denotes a mean quantity, and hence the value associated with the quiescent state. ~m is the temperature at the middle of the fluid layer while pure conduction is taking place, and ~o is the temperature of the upper and lower boundaries of the fluid. The boundazr temperatures will be forced to remain constant regardless of the motion of the fluid0 The symmetry of the
 temperature about the mid-plane of the fluid is a consequence of the assumed constant thermal properties0 This constancy could result from the nature of 'the fluid itself or from the fact that only small temperature variations are being considered. With equations (3) and (5) it is possible to write the mean density as g =o{ -2( b - 42 (k{'" ~Cjbx3-34)} (8) Now consider snmall departures in the values of the temperature, pressure, density, and velocity from those existing during the conduction state0 The va'lues of these quantities will be if a linear per turbation is employed, and the primed quant it ies are these perturbat ions0 The perturbed variables (9) are then substituted into equations (1) and (2) and equations result from which a solution for the perturbations can be obtained. The solution will be examined to determine the conditions under which a perturbated q:uantity can have a steady-state
 value other than zero. The equation resulting from the substitution of (9) into equation (1) is i/ I. -e- (lo) ~~d pxiiP~' (htu + (10) In writing this equation the primed quantities are considered to be small and all products of perturbation quantities can be ignored because their magnitude will be very much smaller in comparison with the other terms in the equation. The result of this simplification is three scalar equations which are linear. By introducing ~o from (9) into equation (3) and by comparison with the expression for p given in (9), one can conclude that (11) Recalling the series expansion for the reciprocal of (l-X) it is possible to write 4 as -5eX0(t %(bX3- X~g ( For the case that the term O is small and the fluid depth b is also small, the second term in the brackets can be neglected in comparison with unity and equation (10) reduces to -10 -
 '~ti = si3((t -?potaxil-(:x-+)3i4 + PAU. (12) The value of ) in the above equation is considered to be constant for the small variation in temperature which will be encountered. The equation of continuity, or mass conservation, is tg=i = (13) and can be written as,x,. (14) if the perturbated variables of (9) are employed and the terms consisting of products of two perturbation quantities are neglected as before. Substitution of equations (8) and (11) for P and, respectively, into equation (14) gives the following result for the divergence of the perturbated velocity field o (X; {(b-2X3)} i- -(6x-3) (15) For the case that O is sufficiently small, X is small in comparison withand 3, if and ~-l-.adI,i n
 U3 are of the same order of magnitude, One can draw the conclusion from (215) that for t imedindependent solutions,, 'is small compared to U3 o The equation gul, o xi (16) can be used in the sense that 9auax < 1 1'U' By introducing the perturbated quantities into equation (2) to account for ' heat conduction, it is possible to write after neglecting the perturbation terms of the second power 4-t UL Xi = /\);i, U/, (17) since dt, and uig " A~ +t from equation (5), Equations (12) and (17) can be used to solve the stability problem if the terms associated with the pressure -12 -
 contained in (12) can be, eliminated. This is achieved by cross-differentiation of the three scalar equat ieons of motion and making appropriate combinations of the terms in conjunction with the continuity equation given in equation (16). Thus, writing out the scalar equations9 one obtains, ruat podx,~ (tti (12b) (12c) By operating on equations (12c) ((12a) (12c) and (12b) with.9..9~:~.9 respectively, one obtains %t u 3,!' I, 9u, -9 ~P;xo~
 (20) a/X~ -Fo i-2X,~ -( +/+ )Q) + 1) x /\ (21) A subtraction of equation (19) from (18) and equation (21) from (20), and then adding the two remaind'ers gives One notes that the terms which aze urnderlined in the above equation have been introduced for convenience and do not change the value of the expressions, Equation (22) reduces to 'dn,,U3_ ut> _.Y- /xs (23) ine notes hat twi Ahet which ine nh4 3~~ r n9 X 9X (31a l~l
 From equat ion (15) it was possible to conclude that %LUy/a3X is an infinitesimal of an order higher than 3 Thus9 while X may not be exactly zero, i.t is assumed that 7 will be small in comparison with Lu3 9 and may be neglected in terms that also contain AU3, The validity of this assumtion will be examined when the solution for the velocity components is obtained0 After neglecting the derivative of the divergence of the perturbed velocity9 equation (23) becomes 2Ai3) =' o(AjO'3UAAUs. (23a) A rearrangement of the above result gives the second of the two desired governing equat ions as (24) The solution of equations (17) and (24) along with the appropr iate boundary conditions - is necessary for the determination of a stability criterion for the flow under study0 The attendant boundary conditions are ~l5S
 (25i) uL=o - at x3= o, b (25 1) - o= atxo, b. (25aiii) The first four of these conditions result from the fact that the upper and lower boundaries are isothermal and r gid,, hence allow no variation of the surface temperatures, and no velocity components perpendicular to the boundary, From the fact that (UA and Ut are zero everywhere on the boundary due to the presence of viscosity, the last two boundary conditions can be obtained with the help of the continuity equation. It is possible to proceed directly with the solution of the problem given by equations (17) and (24) in conjunction with the restrictions of (25)- but it is more convenient, and the results more general, if the equations are non-dimensionalized by setting t=Tb ' (X,) -i xt ' E ' WsU3b TS ~-. (26) -16 -
 By employing these new variables in equations (17) and (24) the following is obtained ( 2 QqV)T = -=4w(1 -Z9) AN (27) and (28) in which 7 and? are now the ILplacian operators with respect to the new coordinates (X, ',) and (X,) respec t ively. The boundary conditions (25) become -T=O at,=o, 1 (29i) w-o a't,o,1i (291i) w -o ig ol <(29iii') For the solution of partial differential equations of the type given by equations (27) and (28)9 the method of separation of variables can'be used, If WV and T are assumed to have the separable form We v (30) 7T fTe (31) in which f is a function of X and 3 only) the functions W and T are dependent only on, and Cl has real -17 -
 and imaginary parts J and G,..I respectively~ For W r r greater than zero the solution (representing the disturbance) will grow exponentially with time, and for Qr. less than zero, the perturbation will decay. If (T is non-zero the perturbation quantities will be oscillatory, Upon substitution of the assumed form of the solutions into equation (27) one has e [i-(f.T +> T+f + - 4t )T in which the subscripts denote partial differentiation in accordance with standard practice0 The above result can be written as f X. f 4A1- Z i' aT1 T (32) in which c. is a dimensionless constant characterizing the mode of the disturbance. This constant is called the "cell number"'0 The last equation yields (33) and W ( ~f( D - a T-, (1-21) 344 -18,T
 in which D= The substitution of the assumed form of the solution, equations (30) and (31), into equation (28) yields, with the aid of, equation (33), dAaZ R2 (35) in wh ich (36) At the threshold of instability it is clear that the solution neither grows nor decays. Hence 'C must be zero. The value of 0 cannot be assigned a priori but in certain problems (5) and (15) it has been shown that at neutral stability 3 must be zero. In these cases the differential system is self-adjointo For a large class of probe lems the governing differential equations are not selfadjoint and it is impossible to show mathematically that 0; is zero without extensive and detailed calculation. It is assumed in those cases for which the value of GT cannot be determined simply, that 0 is zero, when 0' is zero, This is the assumption of the "principle of the exchange of stabilities", ForP the present problem this principle will be invoked and T- is taken as being zero -19 -
 at neutral stability0 Hence the system of ordinary differential equations which must be solved is simplified to (D-a )T )(i~1-2; (38) The boundary conditions for these equations are T:0 ar -oli (39i) A w=O at = O) t (39i (39iii) 2. Method of Solution. The solution of the governing differential equations can be obtained readily by the method of Chandrasekhar. This numerical technique provides a means of solving problems for which there is no simple exact solution. The procedure is straightforward and is not too involved, so that one need not resort to electronic computing devices to obtain the' solution, A brief outline of the method is presented in this section. The thermal boundary conditions (39i) will be satisA fied by a function T- having a Fourier series expansion of the form A o T= A~ $ nw rl:.n, 12;?3.., fl-I (40) -20 &mdash
 Using the above expression in conjunction with equation (38) permits one to solve for W. a a A-IR,A, aaiC, sizE5r> CNJnha +El)caa = R1~ Ari{ C C~ ~(41) in which (4 1 Nr=.r()+ a (42) The velocity boundary conditions (39ii) and (39iii) when applied to equation (41) and its der ivative require that By= O (43) (44) Drn- E, cth a - C, (45) E Traa+_(-l)_ a I <,, '1 (46) Substituting the series equivalents for T and W equations (40) and (41), into equation (37), one gets ~21 -
 0(k )- Ay,,iCr (I sAat ~= L'i.'. &mdash 4 na2 k - (47) If thi's equation is to be true, the coefficients of S/t nrTT on both sides of the equation must be equal for all n. o (Note the terms on the right hand side of the equation could be expressed as an infinite series of sine terms)0 The method for equating these coefficients consists of multiplying both sides of equation (47) by S mrt F and integrating from zero to one9 the range of T o This method is the standard one for determining the coefficients of a Four3ier series. The equation resulting from the prescribed integration is then 11.1n1 D0 0 NnAA n i~ i 5(/n l-I t+ sj Tr s~ Aaady 0 C N 0(2
 Because of the orthogonality of the sine function, the above equation can be written upon integrsation as 'Arl Swmkt a 4a M- T + t Mm ~rin which += ~ il I man s 4 M= O M L mL n s ML 4 in= 1 +o m n details of these integrations are given in Appendix Io) (49') e i 4 l y but ins evn n rilt The detilsoftheentegr ren is iA n
 The non-trivial solution of this system of M1. homogeneous equations with rl terms requires that the determinant of the coefficients of Ar vanish: + M M[~ Ml< =] + I'n-r (-I )%c(a -_amll + __ MrZr t 2 Cm MT.) + 7n Nn (50) The works of Chancdrasekhar (5) and Yih (15) have shown that the solution of the infinite order secular equation can be-closely approximated by evaluating the determinant for a finite fl In essence the series solutions, equations (40) and (41), are truncated after nh terms and the solution is determined from the finite number of terms that are retained~ In order to simplify the subsequent writing of the secular determinants, let I) -Mr ] (51)
 D= J rJT X+ l mT3( ))l+ -6- i1 (52) (53) and _ Xml ~Orr1H =A' f. CrCk+ Dn D+ Er, E7 &mdash &mdash +,8az &mdash l(54) With this notation, equation (50) becomes -_ Amn =O. (55) In his original paper which described the method of solution, Chandrasekhar found that the solution could be approximated closely by solving the f by n determinant (55) for r equal to one! The accuracy of the solution improves as nf increases but in the case of the Benard problem and one considered by Yih (15) the improvement is less than 10 per cent, However, in the present problem the solution for All= 0 indicates that the system is always stable, whereas higher approximations give a finite Rayleigh number as the stability criterion. This is due to the lack of symmetry of the convection so that the first term (associated with All ) is inadequate to describe the convection, The difficulty with using A 1l=O as a,:25 -
 possible solution becomes apparent as the required numerical calculations are carried out, but it follows directly from the definitions of Cn, Cm, Dn, Dm, Ens Em, and Xmno Then A1 be comes A1 and setting this first element of the determinant equal to zero results in an infinite value for IR, for a non-zero wave number a. It can also be shown:that Amn=O for m+ M an even number (hY1.), and for m=-n These results simplify the evaluation of the secular equation and are given in Appendix ILo Therefore, one is forced to consider the solution of a 2 by 2 determinant in order to get a first approximation to the particular problem at hand. To obtain the required solution - the minimum value of [ as a function of a in equation (50) - one assumes values of a and solves for ~. A graphical presentation of these solutions determines the value of the minimum, or critical Rayleigh number. Fig. 2 shows the result of such a series of calculations, The numerical procedure was carried out for a 2 by 2, 3 by 3 and 4 by 4 determinant in order to improve the accuracy of the solution, A summary of the pertinent conclusions are given in Table Io
 FIRST APPROXIMATION CURVE / SECOND APPROXIMATION CURVE THIRD APPROXIMATION POINT -4 3, STABLE REGION 1 0 2,000 4,000 6,000 8,000 -10,00 1I Figure 2. Neutral-stability curve for laminar natural convection in a fluid with homogeneously distributed heat sources contained between horizontal boundaries that are rigid and isothermal.
 T: A.BLE 1! Summary of Co"rt t of _tr,,$-ea4itiy Curve for a Fluid Having Honogeyn.ey Distrit e4 eat ources Contained Btweew Horizontal.Bo,.ries Tt aRe R.gid and Isothe ral a 2.O:.0 2.5 3.0 3.5 4. 4.5 K '(First Mode) First Approxiitionz. 7818 6000 5143 4885 4654 4731 Sec ond Approx 5 20 4 6.84 Third Approition 4669 R '{Fir st Arox4PPr R (eTir. Approx) R ('Third... R (Firs AOtrox.) Q 9967
 3. Temperature and Velocity Distribution in the Fluid LayerB The solution just carried out for the critical Rayleigh number involved the non-dimensional temperature and velocity A A functions T and W, defined by equations (40) and (41)o These functions depend only on the vertical coordinate A A The nature of T and w can be seen by examining Fig~ 3. The curves presented in this Figure were obtained in the following manner: If the series in equation (40) is truncated at ~n=4, one can write four equations involving Al, A29 A3, and A4. These equations are, in the notation of equation (54), A1All+ A2A12+ A3A13+ A4A14= 0 (56) A1A2+ A2A22+ A3A23+ A4A24= 0 (57) A1A31+ A2A32+ A3A33+ A4A34= 0 (58) A1A41+ A2A42+ A3A43+ A4A44 0 (59) This is the set of homogeneous equations that were used to solve for the critical Rayleigh number in the last approximation achieved. The last three equations can be used to solve A2, A3, and A4 in terms of A1o After this is done / A the expressions for w and T will contain only one arbitrary constant, A1, which is the amplitude of the perturbation. The magnitude of Al cannot be determined
 Z b | T/AI -.8 -.4 0.4.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4.8 6 ) O -.2 0.1.2.3.4.5.6.7.8.9 1.0 1.1 A 2 63 w /a RA1(1 Figure 3. The variation of the nondimensional temperature and velocity functions, T and, with the nondimensional vertical coordinate, z.
 if a linear analysis is used. The recent non-linear work of Malkus (13) and Stuart (14) is capable of giving a solution for A1. The coefficients A29 A3, and A4 are A21 A23 A24 A1 A31 A33 A34 A2 = A41 A43 A44 (60) A22 A21 A24 -A1 A32 A31 A34 A3 A= 42 A41 A44 (61) A A22 A23 A21 -A1 A32 A33 A32 A4 A42 A43 A41 I (62) 'and A22 A23 A24 A32 A33 A34 A42 A43 A44 These determinants can be slightly simplified by recalling that A31, A24, and A42are equal to zero. The values for the various Am are obtained by substituting the critical Rayleigh number and the associated critical -31 -
 cell number into equations (44)-(46) and (51)-(54). For this particular problem the cr itica Rayleigh and cell numbers are obtained from Fig. 2 and are 4672 and 4,0, respectively, The solutions to the three equations (60), (61), and (62) are then A2 A1 (1.429777501) (63) A3 = Al (0,2251148661) (64) and A4 A1 (0005253886715) (65) A In view of the above, T, the part of the nondimensional perturbation temperature that is z-dependent, is T= A1[ r(- 1.4Z9777501 + o.ZZ511486 61 S + 0o05Z53886715 347 4 (66) This is the expression that is plotted in Figo 3 for T With the evaluation of the first four AtS in terms of Ai the first four terms in the series for W become -cf. equation (41)
 w -A A[C shA + Dil sfWaa + ElUh3~ +IL &"/rrT _-A[(i. z..)Cs~AaZ +(.4lL.) D{ai + (i.i. )EzDI + A[(o.2-~')C3s/vYwj/~(o.22~z)') )D3' +~ (o.+ )E32) t+ A1 [(o. o5.'3) C4&Srj3' +( o5)D43 $1AAz + (&.050. E4 + (05-Dg56 41 in which only the first two decimal places of the constants given in equations (63), (64), and (65) have been written for the sake of economy. This expression for atR can be rearranged for more effic ient evaluat ion as
 w T Aes+ C,-(1.4z2..)C + (o.z...)C + (oo. o,)c4 + +,(1.4",)T +(0.2)s + ) (0o.05.)-,4jj 1i~~~~~ 2 1 ~(67) in which R = 4672 and a 4 as before, and the coefficients C!, C2, C3, C49 D1, D2, etc., are determined from equations (44), (45), and (46). The evaluation of the above equation for W as a function of Is shown in Fig. 3, The solut ion that has been obtained for W is deficient in one regard, The direction of W at the center of the cell is indeterminate. Whether the flow is upward or downward at the cellts center could be learned from an experiment, For the Benard problem the flow is downward at the center and upward along the sides of the cell. The recently developed non-linear analysis is capable of pr edicting the flow direction for the Benard problem -
 and does so in agreement with the experimental observation' After vW and T have been determined, there remains only the specification of f in equations (30) and (31) to completely describe W and T (up to the undetermined amplitude, A ). The quantity f must satisfy equation (33) which is rewritten here for convenience: tx +f t +a f o (33) This equation is amenable to solution if f is considered to be a separable function of X and?, ioe., j= lAd, =j(X),k=-(aj). (68) (Only the spirit of the solution for I will be given here so that the form of the i that was assumed will be clear). When f, from equation (68) is incorporated in equation (33) one obtains or a~g+t=0) R, &SCX+PZjCmoCX), (69) and (70) Thus -35 -
 The specification of the constants in this equation is achieved by imposing the necessary boundary conditions on i. These conditions can be obtained from a knowledge of the desired velocity field, The conditions on are determined by considering equation (31) in conjunction with equation (16). This latter equation can be rewr itten in non-dimensional form by recalling uAX_ 9xu x + + 3 and = X- - etc L (jb e etc b (< whence dx.g X. At the conditions of neutral stability (ite., -=0 ) w fand equation 33 is valid so that and equation (33) is valid so that XM i4I t1 s Iat io b Wo Multiplying this last equation by, one has 36
 ?v]; +j ~'25 aog " 925 dt(71) which, upon compar ison with equation (16) becomes if (72j) and (72ii) Because there is no coupling of L: ~ V. and W in the equations of motion (10)9 a and V can be chosen in the manner above0 One must prescr ibe the locat ion of the cell boundar ies in the X-~ plane and the required vanishing of the components of WU and \/ perpendicular to those boundaries will create the related restrictions on and so as to specify the function In solving for f the separation constants E1 and C enter into the problem via equations (69) and (70) As a consequence of th$is fact9 a ratio exists betweenl the dimensions of the cell. a 0
 in the X-t plane and the height of the cell in the direction. It seems logical, therefore, to call a the cell number in view of the role it plays in the cell's geometryo The solution of f for the case of a hexagonal tesselation was obtained by Christopherson (16). For this configuration f has the form {{o E t T X + X>+ Xjix + (t4 4fKTFrz1 i^= V [C 3L 1Xf ) + t z] in which fo is a constant, L is the length of one side of the hexagon, and p is an integer. The equation relating v/b and a is Thus 5 can be written as I 4~[em(~x~'i9~ctcb(a 3x1)~&otUj (73) It is this function of f that was utilized to complete the evaluation of W and T in equations (30) and (31). A hexagonal cell shape was specified because that shape was experimentally observed by Benard for the problem bearing his name, However, when experimentally studying the BInard problem for cases in which the fluid layer is confined between solid vertical boundaries that are closely spaced, it is found that the shape of the boundary influences the.a 3 U
 shape of the cell. Thus, for a horizontal region that has a diameter of 8 inches, the cell shape consists of concentric rings; for a square boundary with a side of 6 inches the cell shape consists of many cells which are approximately square in shape. In view of this9 the form of any streamlines that are constructed by assuming a i an f 9 may not have any physical counterpart in a particular experiment, It should be noted9 however9 that the indeterminatecy of f for the present problem, which was treated by a linearizing process9 in no way affects the stability criterion that was obtained~ The streamlines in the vertical.planes of symmetry of the cell can be found with the help of equations (71), (72), and (73)~ For the motion in the plane for which 1=O, V-O because of the symmetryo Consequently, assuming a stream function, 9, such that one obtains,' upon comparison with equation (69), (= W xa O a(I f. O=(743 In the same manner the stream function, -F, which exists in the plane X=O, is -39 -
 1aw (75) Equations (67), (73), and (74) are now employed to determine the streamlines in they = plane: /\ < t ed (X)a (OX' ' t r A whence and for a=& ~ XW v'x G6VTx. (76) The streamlines in the plane of symmetry Q=0 are plotted in Pigo 4 by using equation -(76)o It can be seen from this figure that the mean parabolic temperature distribution has resulted in a flow that is asymmetrical with respect to the midiplane of the fluid layer, whereas in the Benard problem the flow is symmetrical about this plane, 4, Heat Transfer Aspects of the Solution. Once the temperature distribution T is known for * Al n
 -41/ yi SECTION A-A Figure 4. The cell pattern in one plane of symmetry at the onset of convection for the first mode and for an assumed hexagonal tesselation.
 the convection (or unstable) state the temperature gradient can be found along the upper and lower boundaries of the fluid layer. By integrating this temperature gradient over the entire horizontal fluid boundary, it is possible to determine the fraction of the total quantity of heat transferred across each of the two horizontal boundaries0 Hence, to evaluate the quantity of heat flowing across the lower boundary one writes A (~) dA QL- ~h'X3/x o0 (77) in which the integrat ion is carried out over the hexagonal lower boundary of one convection cell, and the subscript on Q denotes this lower boundary, From the definition of ~, the vertical temperature gradient is 9x3 X3 '8)X3 axX b Oa 9a ~ La( ~ + (f) ~, 9XX?X3 b aX (78) The heat transferred, equation (77), can be rewritten as At= ' 8 xldX. + t(~A~o) "f.)/"/|" ' dxd A A in which the first integral represents the amount of heat transferred by the mean temperature distribution - a value
 which because of the symmetry of the temperature distribution is the same at the upper boundary - and the second integral represents the increase (or decrease) in the amount of heat transferred at the boundary due to the perturbation. If the second integral is examined in more detail its magnitude can be obtained without the necessity of a formal evaluation. 'With the aid of equation (33), this integral becomes,ttfY d -T + dx d A A and by means of Green's Theorem the area integral can be written as a line integral such that td =x A C in which the integral on the right hand side is evaluated on the hexagonal closed curve comprising the boundary of the cell in the X-~ plane. The line integral can be given a vector interpretation which is convenient for its evaluation -- cf., Kaplan (17). Consider a vector F=_:i + J _ax in which l and j are base vectors in the X and 't directions, respectively. A unit vector normal to the -43 -
 hexagonal boundary can be written as iwc - s theds J in which S is the distance along the boundary0 Consequently, -L qf _gf dx dX in which Fn is the component of F in the Y1 direction. Then ~i4a -c Xf dX fs -+F ds It will be noted that the vector defined as F is the temperature gradient in the X- plane (the lower fluid boundary for the situation under discussion)0 The integral just obtained requires that the component of the temperature gradient which is perpendicular to the hexagonal boundary be integrated along the boundary. Examination of the dimensionless perturbation temperature, T, shows that, for a prescribed value of? 9 the isotherms are everywhere perpendicular to the hexagonal boundary. Hence, the temperature gradient has no component perpendicular to the boundary and Fn ds-o
 One concludes from this result that there is no net change in the amount of heat transferred at the lower fluid boundary as a result of the onset of convection0 By direct analogy it is apparent that this same con= clus ion holds for the upperl fluid boundary0 The fact that the sum of the heat transferred to both boundaries must remain a constant9 regardless of whether conduction or convection takes place is to be expected9 and required9 since the amount of heat liberated by a unit volume of the fluid in an interval of time is a property of the fluid that is considered to be independent of the type of motion. However9 it seems reasonable that under convectiton condi tions there would be more heat transferred to the upper fluid boundary than the lower boundary if both were kept at the same temperature0 The question can evidentally not be resolved by a linear theory, It can be speculated that a nonlinear analysis might bear out this conjecture. 5 oReiew of Ass&2 A.ons In the course of developing the governing equations a number of assumptions were made which resulted in a set of linear differential equations that were subsequently solved. These assumptions were the following: ao The thermal properties of the fluid would remain constant, despite the presence of small temperature differences in the fluid0
 b. The viscosity of the fluid would remain a constant over the range of temperatures existing in the fluid layer. c. The Laplacian of U3 would be large compared to _a with the result that the latter could be neglected by comparison with the former. d. The conversion of mechanical energy to heat could be neglected in writing the energy equatiaon. The validity of the first of these assumptions can be verified with a list of the physical properties of the fluid in question. Water, which was used in the experimental work, has thermal properties which satisfy this assumption. These are -- cf., Handbook of Chemistry and Physics (18). Temperature (~C) 150 200 Thermal conductivity, k 0.00144 0 00143 (Cal/cm/se c/C) Specific heat, Cp 0.99976 0o99883 (Cal/g/0C) Density, P 0,999099 0.998203 (g/cm3) The coefficient of thermal expansion for water is also constant for small temperature variations, Its value
 At 200 C is 2(10C4)/ Ce The validity of the assumption regarding the constancy of the viscosity is borne out less well. For the temperature range 15G20O C the dynamic and kinematic viscosities vary approximately 12 per cent, However, if the temper= -ature variation in the fluid were about 20 C. the resultant variation in viscosity would have a minor effect on the pr oblemo When considering the third assumption it is convenient to employ VV2 in the calculation of Au3 Thus ox~+ 9,f+;,.:,,',~w + W and from equation (33) A A rA qz ~ Jxxw + w = - w - Theref ore A It follows that from (67) 44a A _. A 1 Nr, A1 N5 A _ ki. j 1 Ai a 71.J AT sNA U U$7
 in which the constants S and T are the coefficients of 5A? and Cat gisven in equat ion (67) eThe above equati on can be written more imply as w 2, + aARA1[P], in which SO TI Az _4 _ A1 N3 A Thus au, =-v(a8RA,) Pi From equation (15) one can write for steady-state conditions and 3 1 It. cT- 2u3. ( b - x,) __ ~48~
 9!ijiL, O1!H r-zw + I-ZVI i8 xg3 Uxi, 2Lbi n'Vt( ~2 To compare the magnitude,of AUi and one can calculate their ratio as tAU, 2 [I A3 LIR ~ The denominator contained in the bracket will be referredto hencefor th as b o Thus QXR(9AL) T, L The function L can be readily computed once is determined' Again using equation (67) one has at'R i4A - +.A\(') =/ aS iQ7i ~. S- d+ aS?. )1\ a 4- r ~;;~li~'6aTt<-/J 4,)4 A Sr (I 4.7 +._ _ _ __' ~ ] u V A
 in which ~ is the coeff ic ent of $ni Ui in equation (67). The ratio P/L was computed for o.~.1 and the results are shown in Fig. 5, By using the physical constants for water, it can be seen that only in the minute regions 0o86< ( 0.87 and 0.288 < 0.291 does the rati~o LA I3 ix fall below the value of 200 In fact, in almost the entire region the ratio exceeds 1000o Hence, it can be concluded that, except for a small region at Q = 0.865 and a smaller region at 0o290, the assumption AjU ~ ( is jus if ied In writing the equation for heat conduction the term representing the viscous dissipation of mechanical energy to heat (Rayleigh Dissipation Function) was omitted, This omission is reasonable in a problem for which the velociP ties and the velocity gradients are small0 Not only is it to be expected that the dissipation function will be small but its effect will be further reduced by multiplying it by the dynamic viscosity and doividing by the Joule constant, 50()
 1 &mdash -&mdash 0. -8 -4 0 4 8 12 16 20 24 28 32 A U3 4a (Om l60) CX ( I) Figure 5. The ratio of the Laplacian of the vertical velocity component to the derivative, in the vertical direction, of the divergence of the velocity.
 CHAPTER III 1MATHEMATICAL FORMUIATION AND SOLUTION OP RELATED STABILITY PROBLEMS The stability of a fluid with homogeneously distributed heat sources that is conf ined between two horizontal boundaries can be studied for boundary conditions other than those used in Chapter II. The mathematical boundary conditions on the velocity and temperature that can be imposed at the upper and lower surfaces of the fluid corresponds to the various physical conditions which can exist at these surfaces. These new boundary conditions will result in different solutions, even though the governing equations are the same or similar, because the solution must satisfy the differential equation and the applicable boundary conditions. In this chapter two additional solutions are obtained for the thermal stability of a heat generating fluid, In treating these two problems the notation and assumptions adopted in Chapter II are retained, 1o Rigid Thermally Conduct ng Upper Boundary and Rig id Therma. Inulating LowerF Boundar >tx\er mally Conduc ting Rigid BoundaryX3\ XI Thermally Insu ating, Rigil Boundary Fig.6 Sechematic Drawixng Of Fluid Layer With The Mean Temperature Distribution And The Coordinate System,
 For the first problem to be considered, Pig.6, the mean temperature is given by h(jbLZ) t~ (79) The method of der iving the governing equations is identical to that previously employed: a. Perturbation variables for Ci 9 ~ 9 p and 0 are introduced into the equations of motion (1) and heat conduct ion (2)X bo All products of per turbation variables and their derivatives are neglected as being small. c. The equations of motion are crossedifferentiated and selectively added and subtracted to eliminate c. The term a ~jj is neglected by comparison with tU3 in conjunction with magnitude considerations arising out of the continuity equationo The governing differential system that results from these operations is 8and (q2; -g t1 () = ($2A 23
 The boundary conditions are L3= o at x3=, b; (82i) %X3 ~ =~o ~t X3- o 3-6x3 (82iii) ~'-o at x3-b. (82iv) The dimensionless variables, (9), can be introduced so that the governing equations are (83) and V7- T ( = Zw(i w=o at -oli 8S1) WA 0 h 9=- i (85 iii) w3~~ o8 -i
 UT=O a (85 iv) The solution of the problem posed by equations (83) and (84) is achieved by the method of separation of var iables so that A _ C w=!we T= TGe in which f is a function of X and / and the functions A A W and are z-dependent. With the velocity and temperature so defined the governing equations and associated boundary conditions are [ D - -a (d ) - m ( )D-aZ T R7D (86) and [- (D )- a = w (87) with w=-0 a ~= O; (88i) ___=O al ]=o, li 3' 9(88ii)
 (88iii) T-c at }1 (88:iv) The boundary condition (88iii) results from the prescription of an insulated lower surface, The "'principle of exchange of stabilities" is employed as before so that 7=C), and the equations which must be solved reduce to - dZ) azaRT (89) and (DL / at)T- zw (90) The solution of the problem stated by (88), (89), and (90) is achieved by the approximative method used in A Chapter II, The assumed form for T is C>) T = A, c-I )3= 1 3)5)' TL (91) allows the thermal boundary conditions to be satisfiedo Equations (89) and (91) are combined to yield
 :,) J=',3,5, (92) in which Z Z The velocity boundary conditions require (93) Cn -- &mdash. (94) D,=-B- (Cn+ E)(cct~a ) ) (95) En= -a B. (96) In order that the two series given by (91) and (92) can show the equality stated in (90) the following secular determinant must be satisfied; -57..
 vBn '+Ct +Dn-D +EnE + )'. D"'+ tJ.(97) in wh ich 3 M r= 9 Xta-i~:(98) c (mT-/i)(-1) I-.+ 45 ] (99) R+3 L + -Y3 4 ( I /c)(-~ a~ + 4d vFl)/z 1 ( 9/a Mm (100) -585 Yn(~0-8
 SECOND APPROXIMATION CURVE 5 FIRST APPROXIMATION 4 CURVE a STABLE REGION 3k UNSTABLE REGION 21 1 i. THIRD APPROXIMATION __ POINT O -. -. -~~~~-. - - - - - I -t 0 ~ 1,000 2,000 3,000 4,000 gcas(Om- e0) b K Y Figure 7. Neutral-stability curve for laminar natural convection in a fluid with homogeneously distributed heat sources contained between a lower adiabatic rigid boundary and an upper rigid isothermal boundary.
 1.0 _ _ _ iii.8.6.i I "II I T/Al N~~~~~~~~~~~~~~~~~~~.4 ~ 0 /~~~~~c.2 i i 0 2.4 6.7 1.0 1.2 1.4 1.6 1.8 A A3 T/AI AND W /o RA1(I ) Figure 8. The variation of the nondimensional temperature and A A velocity functions, T and w, with the nondimensional vertical coordinate, z.
 Xwmn=r i4L (i tor I.n 4c - (. ~-4 for rH- even and m n) - fr n+m odd nd m n) (102) - O -For n+6mTl) (103) and M, &mdash _(mrz ' +, The relationship between a and 1k which must be maintained for a solution of (97) is obtained by solving successively this determinant for m: n = 1, m = n = 3, m = n = 5 etc, The results of such computations are A given in Fig. 7. The functions T and W are plotted in Pig. 8. 2. Fr ee Thermally Condctting Upper Surface and Rigid Thermaly Insul at Lower "ou'ndar y... X Pig, 9 Schematic Drawing Of Fluid Layer With The M-ean Temperature D istribut ion And The Cordinate System.
 The problem described by Fig. 9 can be solved in exactly the same manner as the two previously discussed~ Indeed9 the governing equations are those given by (89) and (90)9 ioeo, I -a v -8RT and The boundary conditions for this particular case are W =0 at ~-' = J0 i (104i) ZA U- 1-~lL)~~~~~~ (104 iii) 2+<. 0 (104iv) (104v) Boundary condition (104iii) results from the free surface which requires that there be no shear presentThe variables T and W have the forms &mdash 62,
 A 7, R w_ at A{ [n t3 1+ C, u-4 + DR n 4a (106) in which ixn=(n/z) + EL '-l= ~n a ~ (107) cn=-'N (108) rn (sLv taE~- a) (109) E= -- a _. he secular determinant which must be sol4ved is / El3+CnCmt D~Dm+ E5 Xmn na~n, n ( in which the Bn, etc., are obtained from (107)-(110) and the B, etc., are those given by (98)-(103). The result of solving this determinant for Y\= 1, 3, and 5 is given in Fig. 10 -63.
 , X - __ __ __ __ ___i, __ _ _ SECONRST APROXIMATION. X~usrA81~ REGIO# 1.... -o a - eoCURVE
 3, Other Related Problems. In Chapter II, one stability problem for a fluid with homogeneously distributed heat sources that is confined between horizontal boundaries is discussed in detail. Two additional problems have been briefly dealt with in the present chapter. These last two problems were the result of imposing different mathematical boundary conditions at the upper and lower surfaces of the fluid layer. These boundary conditions correspond to different physical situations, and additional cases can be studied. A different solution would result from having: a, Conducting free surfaces at the upper and lower boundar ies, be Conducting free surface at the upper boundary and insulated free surface at the lower boundary, c, Conducting free surface at the upper boundary and conducting rigid lower boundary, d. Conducting rigid surface at the upper boundary and conducting free surface at the lower boundary, e. Conducting rigid surface at the upper boundary and insulated free surface at the lower boundary. All of these cases would permit a time-independent, stable state to exist. The solution to these problems could be obtained in the same manner as those treated in this and the preceeding chapters For those cases in which both surfaces are heat conducting, equations (37) and (38)
 would be the governing relationships. The B'S, etc., to be used in (50) would be the same as those listed by equations (51), (52), and (53). These coefficients are listed in Appendix III. The various 3Vs, etc., would come from the particular boundary conditions. The cases for which the lower boundary is insulated could be' studied by using equations (89) and (90). Once again the boundary conditions would determine the E 5, etco, but the E 6, etc., are independent of these conditions and are found also in Appendix III 4, Problems Similar To Those Discussed, A set of problems which are related to those described by Figs. 6 and 9 consists of studying the thermal instability (in the sense of secondary flow) of water at 40 Celsius, At this point the density-temperature relationship has a first derivative of zero which results in the well-known fact that water at 00 C. is less dense than at 40 C. The density-temperature relationship is approximately parabolic in the neighborhood of its maximum point as can be seen from Fig. 11 It is this fact which makes the problems similar to those just studied since the density variation induced in the heat generating fluid is also parabolic. However, one item keeps the two, seemingly similar, situations from being equivalent, For the heat generating fluid there is a term in the heat conduction equation that is not present in the problem of water at 41 0~~~~~~6
 0.999980 96C 940 TABULATED DATA,E 920 ~ 880 / AP ROXIMATION Z 2. w 60- -0. 9 1 9 973) _ 820 - 0.999800__ 0. 2 3 4 5 6 7 8 TEMPERATURE,, C Figure 11. Temperature-density relationship for water near 40C.
 40 C. Consequently, it can be expected that the equations governing the stability of water near its freezing point will be different from those given by (37) and (38) or (89) and (90). Because a discussion of this new stability problem is germain, but not completely relevant, to the stability problems of a heat generating fluid, the solution is given in Appendix IV.68'
 CHAPTER IV EXPERIMENTAL VERIFICATION OF ANALYSIS An experiment was performed to verify the critical Rayleigh number which is a consequence of the analysis in Chapter II, The result of this exper iment is in substantial agreement with that given by the analysis. 1o Desription of Appaoratus The apparatus, shown schematically in Figure 12 consists of the following: a, Two copper plates, 11.5 inches square and 0o5 inches thicko b, Three plastic plugs of predetermined height which separate the copper plates and thereby form the test chamber. co A lower cooling chamber in which cooling water is circulated so as to keep the plate at a constant and uniform temperature, d, An upper cooling chamber which functions as c, above. e. An acrylic plastic housing which contains the copper plates and cooling chambers and permits viewing of the test region. f, Water, having suitable electrical conductivity, placed in the test region between the copper plates,
 THERMOCOUPLE - A.C. CONNECTION COOLANT INLET (SPRAY) COOLANT DISCHARGE STEEL FRAME PLASTIC~ SID COPPER PLATE TEST R PLASTIC SPACER THERMOCOUPLE COOLANT DISCHARGE ] 4 a A.C. CONNECTION COOLANT INLET 12SScheaRA armfep --- a- LEVELING SCREW Figure 12. Schematic diagram of experimental apparatus.
 Also employed in the work were a variable output voltage transformer (Variac), copper cons tantan thermocouples (#36 wire), a potentiometer (Leeds & Northrup, K-2), a voltmeter and an ammetero The copper plates were hand-lapped so that they were smooth and flat to within 0.5 inches of each edge. These edges were slightly rounded as a result of the shearing operation which cut the plates to size. It was the opinion of Assistant Professor Ko Moltrecht, Department of Mechanical Engineering, that any machining which would be extensive enough to result in having the plates flat over their entire surface could result in serious warpage of the cold-rolled plates. Consequently, the plates were hand-lapped only where necessary and the surface was flat over an area 10.5 by 10.5 inches. The fluid layer that existed between the copper plates was heated by passing an alternating current through it by means of the transformer. This transformer allowed the power dissipated in the film, and consequently the temperature distribution, to be variedo The surface temperature of each plate was measured and in the case of the lower plate, at least three thermo.couples were installed so as to be sure that the plate was at a uniform temperature. These thermocouples recorded the temperature at the fluid-plate surface. In addition, the temperature of the fluid at the mid-plane of the layer was measured and also at a point intermediate to the ~71l
 copper surface and the mid-plane. The location of these thermocouples was determined with a depth gauge prior to the experiment0 The thermocouples were made of #36 wire, each strand of which was nylon coated and the two insulated wires jacketed together in nylon. These thermocouples were calibrated against a steam and an ice point. In addition, a test was made to determine the characteristics of the thermocouples between 00 and 300 C. These tests were conducted in the Sohma Laboratory, College of Engineering, University of lichigan. The results of these calibrations were used in interpreting the potentiometer readings during the stability test. Commercially distilled water was used in the test section of the apparatus. This fluid has sufficient electrical conductivity to permit its use and its welltabulated values for viscosity and thermal conductivity made it a desirable medium. The cooling chambers that kept the plates at a constant and uniform temperature contained a spray system so that fine jets of water were directed toward the plates, In this manner the coolant was kept in an agitated state and any dissolved air that came out of solution would be swept away before it had an opportunity to become a large insulating bubble0 The fluid was removred from each cooling chamber at 4 locations which were near the copper plates. -72 -
 This was done to facilitate air removal and to increase the "scrubbing" action of the coolant, In order to observe the fluid motion, prior and subsequent to convection, manometer fluid with a specific gravity of 1.000 at 200 C was injected into the test region. A #25 hypodermic needle was used for this injection, and the small bore of the needle resulted in extremely fine oil particles. Prior to convection, these particles would eventually move to one of the copper plates to which they would attach themselves, This migration resulted from the minute difference in density between the oil and the distilled water in the test sections After the onset of convection the oil particles moved in a regular and continuous flow pattern, In order to observe the particle motion the test chamber was lighted from the back side so as to silhouette the particles 2, Exper imental Procedure Cooling water was allowed to circulate so as to bring the entire apparatus to a uniform temperature. Readings of temperature were taken to confirm this uniformity. A small potential difference was applied across the plates and after a 10 minute interval the thermocouple outputs were recorded~ These readings were repeated 5 minutes later to determine whether
 equilibridm had been achieved. Oil was injected and the particles observed to determine whether they had any persistent and regular motion. If the particles eventually came to rest it was concluded that convection had not commenced. Under these conditions the voltage across the plates (and consequently the power dissipated in heating) was increased a small amount and readings were recorded. This procedure continued until convection was observed. After convection state was established, the power was slowly reduced until the conduction state reappeared. In the course of lowering the power dissipation, thermocouple readings were taken so as to establish that equilibirum conditions (in the mean) were present whenever the nature of the fluid motion was being observed, Several experiments were conducted to verify the analytical work and the fluid depth b was varied as a parameter. It will be recalled that this dimension enters the Rayleigh number to the third power. 3. Results of Experiment The results of the experiment shown in Table II, are in substantial agreement with the analytical work. It can be noted that the experimentally determined critical Rayleigh numbers are higher than predicted. The inability to observe the exact instability point by means of dye or the oil particles and also the
 TABLE II SUIMARY OF EXPERIMENTAL DATA (Data Associated with the Onset of Natural Convection in a Fluid with Homogeneously Distributed Heat Sources which is Confined Between Two Rigid, Isothermal, Horizontal Boundaries ) Plate Spacing, b (Inches) 0 250 0.500 0.745 Fluid Temperature at Mid-plane,L (oF) 69.0 63.8:61o4 Fluid, Temperature at Solid boundary,& (OF) 6505 63~2 61.1 Average Coolant Taemperature (oF) 64 8 61.1 Applied Vol tage Volts 40. 0 24 1 I 0 Current (Amperes) 3.5 1.2 0 4 Kinematic Viscosity (Ft2/Sec) 1o135(i05) 1.17(105') 1o20 (105) Coefficient of Thermal Expansion (1/OF) 1(10-4) 1 (10o-4) 1(o04) ' Thermal Diffusivity (Ft2/sec) (1o55)(10-6) (1o55)(106) (1.55)(10-'6) Rayleigh Number Experimental 5800 7700 12 600 Theore tical 4~672 4672 4672 -"75.
 influence of the vertical boundaries on the instability would tend to give a critical value that was higher than predicted. A more sensitive test for the onset of conduction could be employed by using an apparatus similar to that of Schmidt and Milverton (3)~ This equipment utilized the defraction of a light beam that passed through the fluid, Under conduction conditions the light beam was defracted uniformly across the plate but during convection the light beam was defracted non-uniformily across the plate as a result of the temperature variaiiops that are associated with the convection cell. It is anticipated that the apparatus that was constructed can be used to test and explore other convection phenomena, including the behavior of non-Newtonian fluids. For such experimental worlk a light-technique should be developed and employed. ~76 -
 CHAPTER V RESULTS OF TH0ORETICAL ANALYSIS AND EXPERIMENTAL INVESTI GATION The theoretical analysis for the various cases treated shows that there is a critical Rayleigh number that marks the transition to laminar natural convection from a quiescent state. This Rayleigh number is different for each set of velocity and temperature conditions that can exist at the horizontal fluid boundaries. a. For the case of two rigid and isothermal boundaries the critical Rayleigh number is 4672 and the associated cell number is 4.0. b. If the lower rigid boundary is adiabatic while the upper rigid boundary is isothermal, the critical Rayleigh number is 1393 and the concomitant cell number is 2.5. c. The critical Rayleigh number is 810 and the attendant cell number is 2.25 if the lower boundary is rigid and adiabatic, and the upper boundary is free and isothermal0 Prom the above results one can see that a free surface at the upper boundary of the fluid, case (c), is less stabilizing than a rigid boundary, case (b). It is possible to interpret the results from the case of the heat generating fluid contained between isothermal horizontal boundaries, case (a), in terms of the role played by a free surface on stability. The mid-plane of this layer can be thought of
 as being a free surface, below which there is a fluid with a stable temperature distribution (i.e., positive gradients). Above the mid-plane there is a fluid with an unstable temperature distribution. This layer has a depth of d which is one half of the full distance between the plates, b. At the interface of these two fluids, the mid-plane, neither the velocity nor the temperature is prescribed. By using the half depth, d, as the characteristic length for the upper layer, the criterion for stability of this layer is oneeigth of that found for case (a). The result is a Rayleigh number of 584. Hence, a free upper surface condition is destabilizing and a free lower surface upon which neither the temperature nor its gradient is specified is also destabilizing. The analysis carried out in Chapter III for two problems with non-positive temperature gradient shows that the critical Rayleigh number for those problems is less than 1709, the value associated with the Benard problem (Indeed, this result holds for the problem of Chapter II if our attention is directed to the layer in the top half of the region for which, as was discussed previously, the critical Rayleigh number is 584), Thus, it can be concluded that the mean temperature distribution, and not just the average temperature difference between the upper and lower surfaces, affects the thermal stability of the fluid. From this result one can conjecture that the thermal instability of a fluid which has a
 changing temperature distribution (e.g., transient heating) may not be solely governed by the overall temperature differences incurred. It should be pointed out that the boundary condition on the perturbation temperature in Chapter III is not the same as that for the Benard problem, In the cases which were examined the vertical temperature gradient was required to remain zero (i.e., an adiabatic surface) and in the Benard problem, as well as for all isothermal boundary conditions, the perturbation temperature must remain zero at the boundary. However, a calculation (not presented herein) was performed for the problem described in Fig. 6 but with the artificial thermal boundary condition at the lower surface which required that the perturbation temperature be zero there. This calculation also yielded a critical Rayleigh number which was less than 1709 and thereby substantiated the belief that the higher local temperature gradients associated with a non-linear temperature distribution tend to destabilize the system. For the problem in Chapter II which has a positive mean temperature gradient over the lower half of the region, the cr itical Rayleigh number is increased over that of the Benard problem, if the characteristic length is the full fluid depth, b. Thus, a local positive temperature gradient, favorably placed can have a stabilizing effect on the system. The experimental work described in Chapter IV and sum-79,
 mar ized in Table II is in agreement with the analytical predictions. Therefore, the assumption of the "principle of exchange of stabilities" was justified in the solutions. The cell shape has not been measured.
 CHAPTER VI CONCLUS IONS 1. For the case of a fluid with homogeneously distributed heat sources contained within horizontal boundaries, the onset of laminar natural convection is governed by a critical Rayleigh number, the magnitude of which depends upon the velocity and temperature conditions existing at the horizontal boundaries. A free upper surface is dew stabilizing and a lower surface upon which neither the temperature nor its gradient is specified is also desta. bilizing, A local positive temperature gradient, favorably placed, can be stabil Zing. 2. The Rayleigh number associated with the onset of instability for the fluid under consideration is determined not only by the overall mean temperature difference but also by the mean temperature distribution in the fluid. 3. Detailed relationships between the Rayleigh number and the cell number have been obtained by an approximative method for three problems dealing with a fluid having homogeneously distributed heat sources. The results of these solutions are presented herein via graphs, 4. An experimental study of one of the problems which was treated analytically yielded a Rayleigh number which was an general agreement with the analytical prediction. The result of this experiment justified the use of the "principle..~~~~~~~~~~8m
 of exchange of stabilities" in the mathematical solutions, t aicl sluoios
 APPEND IX I EVALUATION OF INTEGRALS WITH TRIGONOM.TRIC AND HYPERBOLIC INTEGRANDS Note:= C ta ) - ~a~ ' a W (t)Xta1l,' ' ) I _3. Thdd a~'V)(Ob a S+ ' ( ) a6., h{-b' U.t 4\d4l)d-J ~~~ ~~ ~d ~~ iLZ~ib~~h~v~l \ s b I~ZI ~~1~ hi
 KOJCU~~~~~4 cccc ACT~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~j ~4 +,1 0-]:0J -F- t ii f II ~~~~~~>o 0 -C3h! C~U t~or -SQC C -4 01 I ~~~~If 'ES ~ 0 4-CX Dll 0-]f 0.3Tc Q: r-~~cz~.r ~ i _C"I~ ~LQi~W f-~?c &mdash"~~~L)1_~~
 : 5o ~ ii ~~~~~~~(II 01- 04 l.
 Tr nTF r4 4-=r H m even_ L1J _ +or"+n odd, mnr 1 L for m=n-. - for rm-=. I-86I
 APPENDIX II S IMPLIFICATION OF SECUIAR DBETRMINANT F.R SOLUTION OF A PARTICULAR INSTAB1ILITY PROBLBM W,,i \~Ca aE Z4W VV* da vv/s nA a - da,s se - 1 r, E, /E r IF -i$ 4a tsaT 6 a- Arl (iati)( 44a) Hilb~~ I L( ) ' 0, ($041) t 1IJ - j 3 P R t: =,. - 7 + a + 4ar a 4rdT + z7 _HT, + lb U4- a ~ -- a-,: +:,:~ - 1 aa pa ~ 4 da a -albair 4 T,,87.PH
 L11 e, | ts.F ~a ) L~dn) t-ta) = ~tX,.- S ora - 8axrd ~Iar Gd Cti;4RrQ;sr1 Ih L Ll 1;YL~ qX ~ M1~ 'Mtc~ar.aTh ~t E: Mt lMiLBa 112 8t1>M t ETt Q~ ja + Q:oi o H,.113 1 a 1 VLA 'c_ lb~~ 4 J- a r oa 3aU Ea + 4 r scl a =,.-7Lte a + 4ar 8r2aL-$b8 IF I Ah c~a- lW ha - 4 E t 4T4.-~ir - z Zd6 e, rn~ ohd + - 4~v xil aT- 287 Ov~al -- - l4a~ l l,- H ll 4 4r a A&- C 3 a 1 + r+C 8lt ka 4Ta "a AT camp ~~ad
 9 h~it~ CI j- 1.70 i~id-l 3i M<4-i 41,r k LT, ol a Tr A 7i T e i a Fa( X +X)- lla 4S tb1 9Pa + Br 2h7 af &e l 4 &. Mti 8!. ( - (~ 1 s ~t. _ 44 p8 ZIL. IT2~ K ~~~b ~ _~Y b bY s~na'sida~arpaRl+r4Y~~i ii~l rd MI?~~~~ZI QtM~i4 Z~I~ ~~~~P
 Appendix III Coefficients for Approximative Solution to Problems Concerning the Instability of a Fluid with Homogeneously Distributed Heat Sources SOME COEFFICIENTS FOR EQUATION (54) a 2,0 2.5 340 3,5 5.0 7,5 C1 -ol199332868 -0o3168172263 -0.4960498810 -0,7698450147 -2 798851197 -23.44290547 C2 1.007484995 D1. 1859343827 -0O,2886579660 -0o,4428650738 -o,6761169067 -2A 391378528 -19. 97481865 D2 0,7702893738 Et -0O.1891785569 -02 )87920011 -0439758093 -0o 6716829603 -2~ 386097261 -19 97099443 E2 0o7977948156
 Appendix III (continued) SOME COEFFICIENTS FOR EQUATION (97) a 2.0 2.5 3.0 3.5 4.0 4. 1 B 0.31574094 0.45641090 0.65068509 0.92430151 1.31500195 1.87864821 B3 -0L54891307 -o.82347782 -1.22046037 -2.66066941 B 0.64728587 C1 040558217 0.529462)4 0.71075084 0.97422023 1.35691261 1.91417703 C3 -0.60347305 -0.85916531 -1.24298574 -2.66765099 0.66948629 Di 0.20241665 0.29850517 0.43477693 0.63125505 0.91767235 1.3385 D3 -0.41235461 -0.623395456 -0.93167713 -2.06710222 El 0.24062183 0.32790763 0.45763619 0.64920745 0.93191406 1.34994718 E -0.45731265 -0.65474960 -0.95353673 -2.07764530 E5 0.59280352
 SOLUTXION PR E ThERMA L I NSTA3B ILITY OF 'WATE.R AT 40 CE tLSUS /RIGID::H $OH FOZ _7'AI/S Smy$T6MgM r@-4v - -w (c-S~v (.a FIG. $/) (-4)2 Cu (=/ (o)( Y QY. 999979 gm/c.' ~4^N=0 77-/A' O A V2.C 4 Z-48Qdt. VrA/Q7 S A MOAW Q.4evr/,y SUQEJQA/7ue S4 Or PRaMOD $YAf$,S4$ W~/44 6E/56V47W A P~"A1sow7", 4pAYo7.ry,/ 444-.W,-E' wr' 777.E N7,4'AC7aT/ OF CWAPM7' 7. '"rAV' LPo*4 /] x 77?4E /N79040U~CT/O,/ OF "rN4g Pe4,VrQBt. 'v#ep,/4'a5S /'Aro r'4CE /OaL vt~.7OMC' P~:T4'V'lAr/4 7~M4 /4 -"/D.
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 K a _Vz) I:' Wy /G: A A 3 A A A /,.~O A C>+ R'i aw=? wo,., a= oy I i.o, O: A
 =ZArw ) rign~~~~~~~.n 4 = - 2a2 t A, Bn nraW r7 C t2,';'r. -7-E 14EA-r CONUI'r/ON' E)QA17O/ &mdash QO /, 7-A' 'Z q - A.._N /t AZA n + c, S-. -Cn = vn) ~ ~ N N 3 M~i_-r/9t1v/ @SoT-I SIoj! Q,5 -07/5 ~-QU44770N 6' 41. A' W.N W9*C/4 Anlt $ = E~l t;a - ( all _ /,( eA 'r.a )... ~ - ~nv T 4e + 22ai4 Tr naiael. -.
 Cl -( )( (Hr J') ) C'n= _m T1} )'l i n ~" r I" - ~nd SN 1/4) AN;'D TF( nnt)4$ r'ldv5DWSM/%/2> 0/v/E ooEA7 rs/SC/G,7w 4e1as -rckr &mdash " 170/ AT a-3. o. ~4 NOA~ ApP oK.,/AI4/Q S v/s. 7" o8",',Se 7, 'tr'4 Tris..4L,,, - 96.
 B IBLIOGRAPHY 10 Pellewo A. and Southwell9 R. V., "On Maintained Convective Motion in A Fluid Heated From Belows1" Proco Royv Soc, (London)9 Series A 1769 19409 pp0 31s>.3-43 2, Chandcra, Ko "Instability of Fluids Heated From Below." Proc0 Roy. SOC, (London), Series A9 1649 19389 ppo 231 24 2 3o Schmidt9 R. J.o and Milverton9 So Wo9 "On the Instability of a Fluid When Heated From Below9 Proco Roy. Soc0 (London), Series A. 152, 19359 ppB 586-594o 4o Taylor9 G'o 109 "Stability of a Viscous Liquid Contained Between Two Rotating Cylinders," Philo Transo9 Series A9 2239 19239 ppo 289-243o 5. Chandrasekhar S., "'The Stability of Viscous Flow Between Rotating Cylinders," Mathematika. I 19549 ppo 5413o 6. Re^id9 W. Ho, and Harris9 D, L., "Some Further Results on the Benard Problem," Phys. of Fluids 1, No. 29 1958o 7, Lin, Co Co The Theory of Hydrodynamic Stability9 Cambr idge University Press9 1955. 8,1 Ostrach, S., '"Convection Phenomena in Fluids Heated From Below9" ASME Pa er No, 55-A-88.9 Novo 195 90 EckertS EB R. G, Har tnett, Jo Po, and To Fo Irvine, JLo 'Healt Transfer's 'Industrial and En ing Chemistry, Vol. 519 po 453, Maro 19 59 10o Cowling, To G9o M agnetohydrodynamics, Interscience Publishers, Inc., New York, 1957. 11o Nisbet, I. C, T,,, "Interfacial Instability of Fluids of Arbitrary Electrical Conductivity in Uniform Magnetic Fields8" Cornell University9 Ithaca, No Yaop 19589 Office of Naval Research Contract Report. 12o Yih, Co$S,, "Effect of Gravitational or Electro4 magnetic Fields on Fluid Motion,'- Quarterly of Aplied Mathematics, Vol., XVI, No. 49 Jan0 1959 13, Malkus9 WO Vo Ro, and Go Vetonis, "Finite Amplitude Cellular Convrection," Journal of Fluid IMiechanics Volo 4, Part lJsuly 1958, pp. 22526l,
 14, Stuart, J. T., "On the Non-linear Mechanics of Hydrodynamics Stability," Journal of Fluid Mechanics, Vol:. 4, Part 1, May 1958 pp. 1-22. 15. Yih, C.:.-S, " Thermal Instability of Viscous Fluids, Quarterly of Applied Mathematics, Vol. XVII, No, 1, April 19 16. Christopherson, D. G., "Note on the Vibration of Membranes, Quarterly Journal of Mathematics, Oxford Series, Vol. I, No, 4, Ma:r. 94 - 665. 170 Kaplan, W4., Adveanced Calculus, Addison-Wesley Pubishing Company, Inc., Cambridge, Mass. 1953. 18. Hodgman, Charles D., Weast, Robert C., and eSelby, $:Samuel M. (Editors), Handbook of Chemistry alnd.- Phsi cs-, 37th Edi-tion,, Chemical Rubber Publishinig Company, Cleveland, Ohio, 1955. -98 -
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