Heated from Below and Laminar Flow Between

Modified Theories of Thermal Convection in a Layer of Fluid Heated From Below and Laminar Flow Between Two Coaxial Rotating Cylinders

SIAVASH H. SOHRAB Robert McCormick School of Engineering and Applied Science Department of Mechanical Engineering Northwestern University, Evanston, Illinois 60208 UNITED STATES OF AMERICA

Abstract:- Scale-invariant forms of mass, energy, and linear momentum conservation equations in chemically reactive fields are described. The modified equation of motion is then solved for the problems of viscous flow between two coaxial rotating cylinders and thermal convection in a fluid between two coaxial stationary cylinders with different surface temperatures in the presence of radial gravitational field. Also, a modified theory of the Rayleigh-Bénard problem of thermal convection in a layer of fluid heated from below is presented with predictions that are in accordance with the classical results.

Key-Words: - Rayleigh-Bénard flows. Taylor cells. Gravito-thermal instabilities. 1 Introduction

The universality of turbulent phenomena from ρb= n b m b = m b f b du b  u = v (1) stochastic quantum fields to classical hydrodynamic   fields resulted in recent introduction of a scale- -1  w = v (2) invariant model of statistical mechanics and its v= r m u f d u    application to the field of thermodynamics [4]. The b b b b b b implications of the model to the study of transport Also, the invariant definitions of the peculiar and phenomena and invariant forms of conservation  equations have also been addressed [5, 6]. In the the diffusion velocities are given as [4] present study, the modified equation of motion is solved for the problem of laminar flow of viscous  Vb= u b - v b  Vb= v b - w b = V b+1 3 fluid between two coaxial rotating cylinders leading to stationary Taylor cells. Next, the problem of Next, following the classical methods [1-3], the thermal convection in fluid between two stationary scale-invariant forms of mass, thermal energy, and cylinders with different surface temperatures in the linear momentum conservation equations at scale  presence of radial gravitational field is solved and are given as [5, 6] shown to result in stationary toroidal vortices similar to Taylor cells. Finally, a modified theory of the ρ Rayleigh-Bénard problem of thermal convection in a b +�(ρ v ) = W (4) layer of fluid heated from below is presented and its t b b b predictions are shown to be in accordance with the classical results. 秂 b +�(e v ) = 0 (5) t b b 2 Invariant Forms of Conservation Equations for Chemically Reactive p b +�( p v ) = 0 (6) Fields t b b Following the classical methods [1-3], the invariant involving the volumetric density of thermal energy definitions of the density  , and the velocity of  eb = ρ b h b and linear momentum pb= ρ b v b . Also, atom u, element v, and system w at the scale  are given as [4] 2

The conservation of momentum in an arbitrary Wb is the chemical reaction rate and hb is the absolute enthalpy [5]. volume V of fluid with the boundary  and unit outward normal n will be The local velocity v in (1)-(3) is expressed as the  sum of convective w = v > and diffusive velocities   pdV = - pv . n dA + Pij . n dA t 蝌 [5] G G G + rgdV v w  V   V  D ln(  ) a (12)   g g   G that by Gauss' theorem leads to vb= w b + V btg  Vbtg = -k bln( e b ) b p vb= w b + V bhg  Vbhg = -n bln( p b ) c (13) [ +��(pv) - Pij - r g ]d V = 0 t where (V , V V ) are respectively the g tg, hg and hence diffusive, the thermo-diffusive, and the linear hydro- diffusive velocities. For unity Schmidt and Prandtl pb + � (pb v b) = -pρ b + b g b (14) numbers Scb= Pr b = n b / D b = n b / a b = 1 , one may t express Next, the local velocity v in (4), (5) and (14) is replaced by the sum of convective and diffusive Vtg V  g  V  t V = -k ln(h )  bt b b a velocities through substitutions from (7a)-(7c). This

substitution means that quantities (rb , eb ,p b ) could Vbhg= V b g + V b h  Vbh = -n bln( v b ) b cross the boundary  in (12) by both convection as that involve the thermal Vbt , and linear well as diffusion. Neglecting cross-diffusion terms and assuming constant transport coefficients with hydrodynamic Vbh diffusion velocities [5]Since for Scb= Pr b = 1 result in [5, 6] an ideal gas hb= cp b T b , when cpb is constant and

ρb T= Tb , Eq.(8a) reduces to the Fourier law of heat 2 + wb. ρ b- D b� ρ b W b  conduction t qb= r bhΤ b V bt = -c b  轾ρb 2 hρb犏 D + w ρ b. b- b b 臌 t where cb and κb= c b /(ρ b cp b ) are the thermal conductivity and diffusivity. Similarly, (8b) may be 轾hb 2  +ρ + w . h - a � h 0  identified as the shear stress associated with b犏 b b b b 臌 t diffusional flux of linear momentum and expressed by the generalized Newton law of viscosity [5] 轾ρ vb + w . ρ- D2 ρ τ=ρ v V = -m 抖 v / x b犏 b b b b ijβ β jβ ijβhb jβ i  臌 t In the presence of gravitational field, the momentum 轾v conservation equation (6) must be modified to b 2  +ρb犏 + w b. v b - ν b v b  account for a volumetric body force g that acts on 臌 t the fluid even in the absence of motions, and results  = - pρ+ g  in hydrostatic stress within the fluid. Since viscous b b b stresses are induced by diffusional flux of The first and second parts of Eqs.(16)-(17), momentum (10) they will be absent in stationary respectively correspond to the gravitational versus fluid. Hence, hydrostatic stresses will be normal and the inertial contributions to the change in energy expressed as and momentum density of the field. Substitutions from (15) into (16)-(17) give the invariant forms of Pijδ= -p ij  conservation equations [6] in the presence of external force field 3

ρ coordinates, with negligible convective velocities wr b + w . ρ- D�2 ρ W   t b b b b b = wz = 0 and in the absence of chemical reactions  = 0, when radius of the inner cylinder is large as 禬Tb2 h b b + w . T- k� T -  compared to the spacing between the two cylinders tρ c b b b b bp b d= r2 - r 1= r 1 reduce to 禬vp v b + w. v-ν �2 v - + b - g b b v w v 1 p tρb b ρ b b r -ν� 2 v =q q - (21) b b 抖tr rρ r   An important feature of the modified equation of v w v q -ν� 2 v = q r 0 (22) motion (20) is that it involves a convective velocity tq r w that is different from the local fluid velocity v. Therefore, contrary to the classical equation of vz  2 1 p -ν� vz - - g (23) motion, when convective velocity w vanishes, the 抖tρ z entire equation of motion does not vanish but rather 抖v v it reduces to non-homogeneous diffusion equation. z + r = 0 (24) The last term of (20) corresponds to the generation 抖z r (annihilation) of linear momentum due to exothermic (endothermic) chemical reactions [5]. where the reduced Laplacian operator is 抖2 2 3 Modified Theory of Laminar Flow � 2 + (25) 抖z2 r 2 Between Two Rotating Coaxial Cylinders When radial velocity is absent vr = 0, there will be As examples of exact solutions of the modified no diffusion of azimuthal momentum such that equation of motion (20), the classical Blasius wb= v b and the steady form of (22) becomes problem [2] of laminar flow over a flat plate [8],  2 laminar boundary layer flow adjacent to an � wq 0 (26) axisymmetric stagnation-point [9], laminar free convection on vertical hot plate [10], and laminar with the solution axi-symmetric and two-dimensional jets [11] have B B been investigated. In this section, the solution of the w= W r = Ar + , W =A + (27) q 2 modified equation of motion (20) for the classical r r problem of laminar flow between two rotating where coaxial cylinders [12-15] will be investigated. The 2 2 2 2 W2r 2 - W 1 r 1 (W1 - W 2 )r 1 r 2 Couette flow of viscous fluid is considered between A = 2 2 , B = 2 2 (28) r2- r 1 r2- r 1 two coaxial cylinders of radii r1 and r2 that rotate One notes that besides the gravitational acceleration, with angular frequencies W1 and W2 as schematically shown in Fig.1. the stress field in a rotating fluid is also induced by the presence of centrifugal acceleration that tends to z push the confined fluid outward in the radial direction. The pressure is thus expanded as

p= po (z) + p 1 (r,z) (29)

r since in absence of radial flows the equality w = v holds and stresses due to gravitational and centrifugal accelerations lead to r 1 r 2 1 p 1 p w 2 -o = g , o1 = q (30) Fig.1 Taylor cells between two coaxial rotating r z r r r cylinders. The conservation equations (20) and (18) for an The steady forms of (21)-(24) reduce to incompressible fluid in axi-symmetric cylindrical 4

2  1 p1  n� v W 2 = v (31) Tc =1708 (43) r q q r r the flow field forms a series of stationary Taylor cells  2  with the characteristic length a 3.1, as n� vq 2Avr (32) ≈ schematically shown in Fig.1. 2 1 p The comparison between the classical result and n� v 1 (33) z r z the present theory is now discussed in terms of the parameter 抖v v z+ r = 0 (34)  = 2 /1 (44) 抖z r When  = 1, one gets from (35) W = W and Following Acheson [13], in (31)-(32) the azimuthal 1 A = A such that (42) reduces to the classical convective velocity w q is approximated in terms of definition of Taylor number [12] the mean angular frequency  as 4AW d4 W + W T = - 1 (45) w= 2 W r = 2Ar , W = 1 1 (35) 2 q 2 n  and by (3.25) since when W2 = 0 , W = W1 / 2 such that at r= r1 ,  wq = W1 r 1 . By eliminating the pressure term Tc= T c =1708 for  = 1 (46) between (31) and (33) and using (34) one obtains in accordance with classical results [12]. For general 2v values of , the classical results [12] suggest that the n蜒2(  2 v ) = - 2 W  q (36) z r2 average quantity AW in (42) should be expressed as In accordance with the classical results [12, 13] one considers stationary solutions in the forms 1μ+ A W = A W (47) 1 2 vr= v r (r)cos(k z z) (37) such that (42) becomes vq= v q (r)cos(kz z) (38) 4 骣 2 4AW1 d where kz is the axial wave number. By substitutions 琪 Tc = - 2 (48) from (37)-(38) into (36) and (32) and elimination of 桫1+ m n vq between the resulting equations one obtains that by (43) leads to 2 2 2  vr= Dv r = (D - a ) v r = 0 (39) Tc=2 T c = 3416 for  = 0 (49) subject to the boundary conditions for two rigid walls again in accordance with the classical result [12] It is emphasized that the above results were not 2 2 3 2 (D- a ) vr = -T a v r ,  =  , (40) obtained by linear stability analysis of normal where modes as in the classical studies [12-15]. Instead, according to the present theory, stationary cellular d D = , V =(r - r ) / d , a = k d (41) flow structures arise from direct solution of the dV 1 z steady form of the modified equation of motion and the Taylor number has been defined as (20). That is, for any given value of T when Tt > T > Tc (where Tt corresponds to onset of turbulence) 4A W d4 there will be a stationary cellular flow structures T = - (42) n2 with cell size given in Fig.2 of Chandrasekhar [12]. Equations (39)-(40) are identical to the results The problem of viscous fluid around a rotating obtained for the classical problem of thermal cylinder was discussed by Newton who stated in convection [12, 13]. Therefore, when the Taylor Book II of the Principia (1687) [13]: number exceeds the critical value PROPOSITION LI. THEOREM XXXIX 5

If a solid cylinder infinitely long, in an uniform l l and infinite fluid, revolves with an uniform T = = ( )r (51) wW r2 motion about an axis given in position, and the q 1 1 fluid be forced around by only this impulse of the cylinder, and every part of the fluid that is in accordance with Newton's proposition continues uniformly in its motion: I say, that XXXIX. In view of the central importance of the the periodic times of the parts of the fluid are as rotating bucket problem, it is most likely that their distances from the axis of the cylinder. Newton arrived at the above proposition by direct experimental observation. That is, the travel times The above proposition has been criticized by of small pebbles of wood placed in a large circular Acheson [13] who states that : tank of water with a coaxial rotating cylinder could have been readily measured by Newton. Of course, " ..., and it is immediately followed by a proposition that is false; the final statement if one measures T for a variable wavelength of = 2r, then one will find that for complete periodic implies that u is independent of r, whereas the T correct conclusion, on the basis of Newton's orbits vary as the square of the distance from the own hypothesis, is that u is inversely axis of cylinder. proportional to r . ... " 4 Thermal Convection in Fluid It is suggested here that what Newton states in the Between Hot and Cold Coaxial above proposition could also be viewed from a different perspective than that considered by Cylinders in the Presence of Radial Acheson [13]. For instance, it is reasonable to Gravitational Field suspect that the statement : " .. the periodic times In this section solution of the modified form of the of the parts of the fluid..." refers to the different equation of motion for the classical problem of periods of time needed by the fluid at different radii thermal convection in a fluid heated from below is to travel a given distance along its circular path as considered [12-13, 16-18]. The similarities between schematically shown in Fig.2. the thermal convection problem and the problem of The solution of (26) for infinite viscous fluid Taylor cells formed between coaxial rotating outside a cylinder of radius r rotating with angular cylinders discussed in the previous section are known 1 [12, 13]. In order to fully reveal the close frequency W1 is correspondence between these problems, we will consider that gravitational acceleration is in the W r2  r2 negative radial direction of the cylindrical coordinate w= W r = 1 1 ,   1 1 (50) θ r r2 system. This model, that has direct relevance at geophysical and astrophysical scales, is here being ROTATING CYLINDER considered for its ability to better reveal the mathematical correspondence between the two physically diverse problems.  Hence, we consider a viscous fluid between two r2 1 stationary cylinders with the inner cylinder being extremely massive, thus inducing a uniform radial

gravitational acceleration g = (g = g, 0, 0) towards r  r 1 its axis. The walls of the inner and outer cylinder are assumed to be kept at constant temperature of To and v Tt with the inner surface being hotter than the outer one To > Tt as shown in Fig.3. Therefore, the Fig.2 Periodic times of motion through buoyant forces will tend to move the hot lighter fluid distance  at various radii within infinite near the inner surface radially outward, while the viscous fluid around a rotating cylinder. cold heavier fluid near the outer cylinder will tend to move "fall", towards the surface of the inner Therefore, the periodic times T for motion of parts of the fluid through a given circular distance of cylinder. wavelength  (Fig.2) will be 6

z HOT term COLD 1 p g = -g (60) ρo r r Substitutions from (57)-(60) in the steady forms of (52)-(57) results in

r1  2 1 p1 r2 ν� vr - αgτ (61) ρo r Fig.3 Thermal convection in fluid between  2 1 p1 inner (hot) and outer (cold) coaxial cylinders ν� vz (62) in the presence of radial gravitational ρo z acceleration.  2 κ� τ- βvr (63) Since the convection velocity may be considered as v  v negligible, the conservation equations for momentum z r  0 (64) (20), energy (19) and mass (18) in cylindrical z  r coordinates assuming that the radius of inner cylinder By deleting the pressure terms between (61) and (62) is r1? d = r 2 - r 1 , and in the absence of reactions  = one obtains 0, reduce to v  v   2 (r  z )   g (65) vr  2 1 p z  r  z -ν� vr - - g (52) 抖tρ r Taking the derivative of (65) with respect to z and using (64) leads to

vz  2 1 p 2 -ν� v - (53) 2 2   抖tρ z z (   v )   g (66) r z2 τ - κ� 2 τ βv (54) Following the classical results [12, 13] one t r introduces the solutions 抖v v v v (r)cos(k z) z+ r = 0 (55) r r z (67) 抖z r   (r)cos(kz z) (68) where the reduced Laplacian operator  2 has been defined in (25). Substituting from (67)-(68) into (66) and (63) and Following the classical methods [12, 13], the linear eliminating  from the resulting equations leads to temperature profile is given by 2 2 3 2 (D a ) v  R a v (69) T- T r r T T  (r  r ) , b = o t (56) o 1 d subject to the boundary conditions for rigid walls where d = r2 - r1 and is the adverse temperature 2 2 2 vr Dv r  (D  a ) v r  0  = 0,1 (70) gradient. The perturbation of temperature from the linear profile (57) is expressed as where T= Tβ(r - r - ) τ + (57) d o 1 D  ,  (r  r ) / d , a = k d (71) d 1 z and the fluid density is and the Rayleigh number is defined as ρ= ρo [1 + α(T - T o )]  (58) g  d4 where  is the coefficient of volume expansion. R  (72) With the introduction of pressure expansion  The problem (69)-(70) is known [12, 13] to result in p po (r)  p 1 (r,z) (59) stationary cellular flow structures when the Rayleigh the hydrostatic pressure is balanced by buoyancy number is greater than the critical value R c = 1708. 7

The results (69)-(70) are identical to (39)-(40) of where d is the spacing between the walls, is the the previous section for flow between coaxial adverse temperature gradient, and  is the coefficient rotating cylinders. The mathematica l identity of volume expansion. With the introduction of between these two physically diverse problems is a pressure expansion manifestation of the Correspondence Principle of Einstein in the General Theory of Relativity p= po (z) + p 1 (x, y,z) (81) according to which uniformly accelerative fields and homogeneous gravitational fields are locally the hydrostatic pressure is balanced by buoyancy equivalent. The choice of radial gravitational field in term as cylindrical coordinates made here helps to further 1 po illuminate the exact nature of this correspondence. = -g (82) ρo z 5 A Modified Theory of Thermal Substitutions from (78)-(82) in the steady forms of Convection in a Layer of Fluid Heated (73)-(76) result in From Below in the Presence of 2 1 p n� v -1 at g (83) Gravitational Field z r z In this section the solution of the modified form of o the equation of motion (20) is obtained for the 2 1 p1 classical problem of thermal convection in a fluid n� vx (84) heated from below [12-13, 16-18]. Now, one ro x considers a viscous fluid between two stationary 2 1 p1 walls in the presence of uniform downward n� vy (85) gravitational acceleration when the temperature of ro y the lower and upper walls are kept at constant values 2 of To and Tt and the lower wall is hotter To > Tt . k裻 = -bvz (86) Since the convection velocity may be considered By eliminating the pressure terms between (83), (84), as negligible, the conservation equations for and (85) one obtains momentum (20), energy (19) and mass (18) in the absence of reactions  = 0 reduce to 抖v v τ ν�2 (z= x ) - αg (87) vz 2 1 p 抖x z x  vz    g (73) t   z 2 v vy τ ν� (z = ) - αg (88) v 2 1 p 抖y z y x  v   (74) tx   x Taking derivatives of (87) and (88) with respect to x vy 2 1 p and y and adding the results leads to  vy   (75) t   y 2 抖2v 2 v 2 v v T ν�2 (z- z - x y )  2T   v (76) 抖x2 y 2 抖 x z 抖 y z t z 抖vv v 抖2τ 2 τ x+ y + z = 0 (77) = -αg( + ) (89) 抖x y z 抖x2 y 2 The linear temperature profile, temperature By the continuity equation (77), the above equation perturbations, and fluid density follow the classical reduces to results [12] and are given by 抖2τ 2 τ ν蜒2 ( 2 v )= - αg( + ) (90) T- T z 抖x2 y 2 T= Tβz - , β = o t (78) o d Following the classical methods [12], one assumes the stationary periodic solutions such as rectangular T= Toβz - τ + (79) cells of size ( Lx , Ly ) in the forms ρ= ρo[ 1 + α(T - T o )] (80) 8

DW 2p x 2 p y between two stationary cylinders with different vx= - k x sin( )cos( ) (91) surface temperatures in the presence of radial a2 L L x y gravitational field was solved and shown to be DW 2p x 2 p y mathematically equivalent to the problem of Taylor vy= - k y cos( )sin( ) (92) a2 L L cells between rotating cylinders. Finally, a modified x y theory of the Rayleigh-Bénard problem of thermal 2p x 2 p y convection in a layer of fluid heated from below was v= W(z)cos( )cos( ) (93) z L L presented and its predictions were shown to be in x y agreement with the results of classical theory. It was 2p x 2 p y emphasized that according to the present theories, t = t(z)cos( )cos( ) (94) stationary Bénard and Taylor cells are obtained as Lx L y direct solution of the steady modified form of the where equation of motion without invoking linear stability 2p 2p D = d k = k = analysis of normal modes as is done in the classical d , = z/d , x , y Lx Ly theories. Substitutions from (91)-(94) into (90) and (86) result References: in [1] de Groot, R. S., and Mazur, P., n(D2 - a 2 ) 2 W = -ag a 2 d 2 t (95) Nonequilibrium Thermodynamics, North- Holland, 1962. k(D2 - a 2 ) t = -b d 2 W (96) [2] Schlichting, H., Boundary-Layer Theory, McGraw Hill, New York, 1968. where [3] Williams, F. A., Combustion Theory, 2nd Ed.,

2 2 轾1 1 Addison-Wesley, New York, 1985. a = kd , k= 4 p犏 + (97) L2 L 2 [4] Sohrab, S. H., A scale-invariant model of 臌犏 x y statistical mechanics and modified forms of the Eliminating  between (95)-(96) gives first and the second laws of thermodynamics. Rev. Gén. Therm. 38, 845-854 (1999). (D2- a 2 ) 3 W = -R a 2 W (98) [5] Sohrab, S. H., Transport phenomena and conservation equations for multi-component subject to the boundary conditions for rigid walls chemically-reactive ideal gas mixtures. 2 2 2 Proceeding of the 31st ASME National Heat W= DW = (D - a ) W = 0 ,  = 0, 1 (99) Transfer Conference, HTD-Vol. 328, 37-60 (1996). where R is the Rayleigh number already defined in [6] Sohrab, S. H., Scale-invariant forms of (71). conservation equations in reactive fields and a The results (98)-(99) are identical to those modified hydro-thermo-diffusive theory of obtained in classical investigations by application of laminar flames. Proceeding of the the linear stability analysis [12-13, 16-18]. It is noted International Workshop on Unsteady again that the present theory does not involve linear Combustion and Interior Ballistics. June 26- stability analysis of small harmonic perturbations 30, 2000, Saint Petersburg, Russia. imposed on basic solutions. Rather, according to the [8] Sohrab, S. H., Modified form of the equation theory presented herein stationary cellular flow of motion and its solution for laminar flow structures represent local fluid velocity field obtained over a flat plate and through circular pipes by direct solutions of the steady form of the modified and modified Helmholtz vorticity equation. equations of motion (20). Eastern States Section Meeting, The Combustion Institute, October 10-13, 1999, 6 Concluding Remarks North Carolina State University, Raleigh, The scale-invariant forms of the conservation North Carolina. equations for mass, energy, and momentum in the presence of external force fields were described. The [9] Sohrab, S. H., A modified theory of axi- modified equation of motion was then solved for the symmetric stagnation-point laminar boundary classical problem of laminar flow of viscous fluid layer flow. Western States Section Meeting, between two coaxial rotating cylinders. Also, the The Combustion Institute, October 2001, Salt problem of thermal convection in viscous fluid Lake City, Utah. 9

[10] Sohrab, S. H., A modified theory of laminar [14] Lord Rayleigh, On the dynamics of revolving boundary layer flow by natural convection on fluids. Scientific Papers, 6, 447-53, a vertical hot plate. Eastern States Section Cambridge, England, 1920. Meeting, The Combustion Institute, [15] Taylor, G. I., Stability of a viscous liquid December 2001, Hilton Head Island, South contained between two rotating cylinders. Carolina. Phil. Trans. Roy. Soc. (London) A, 223, 289- [11] Sohrab, S. H., Modified theories of axi- 343 (1923). symmetric and two-dimensional laminar jets. [16] Bénard, H., Les Tourbillonns cellulaires dans Efficiency, Costs, Optimization, Simulation une nappe liquide, Revue Générale des and Environmental Aspects of Energy Sciences Pures et Appliquées, 11, 1261-71 and Systems (ECOS2002), July 3-5, 2002, Berlin, 1309-28 (1900). Germany. [17] Lord Rayleigh, On convection currents in a [12] Chandrasekhar, S., Hydrodynamic and horizontal layer of fluid when the higher Hydromagnetic Stability. Dover, New York, temperature is on the under side. Phil. Mag. 1961. 32, 529-46 (1916). [13] Acheson, D. J., Elementary Fluid Dynamics, [18] Jeffreys, H., The stability of a layer of fluid Clarendon Press, Oxford, 1990. heated below. Phil. Mag.2, 833-44 (1926).