The Energy Equation +

APPENDIX A: Conservation of Energy: The Energy Equation 507

APPENDIX A

CONSERVATION OF ENERGY: THE ENERGY EQUATION

The derivation of energy equation (2.15) in y dz Section 2.6 is presented in detail. We dy consider the element dxdydz in Fig. A.1 dx and apply conservation of energy (first law of thermodynamics). We assume contin- x uum and neglect nuclear, electromagnetic z and radiation energy transfer. Our starting Fig. A.1 point is equation (2.14) [1]:

A B Rate of change of Net rate of internal and kinetic internal and kinetic energy transport by convection energy of element (2.14) C _ D + Net rate of heat added Net rate of work done by by conduction element on surroundings

Note that net rate in equation (2.14) refers to rate of energy added minus rate of energy removed. We will formulate expressions for each term in equation (2.14).

(1) A = Rate of change of internal and kinetic energy of element The material inside the element has internal and kinetic energy. Let uˆ internal energy per unit mass V magnitude of velocity Thus w A >@U (uˆ V 2 / 2) dxdydz . (A-1) wt 508 APPENDIX A. Conservation of Energy: The Energy Equation

(2) B = Net rate of internal and kinetic energy transport by convection w (uˆ V 2 / 2)U w dxdy ( uˆ V 2 / 2)U w@ dxdydz wz >

(uˆ V 2 / 2)Uu dydz (uˆ V 2 / 2)Uu dydz

dy w 2 (uˆ V / 2)Uu@dxdydz wx> dz dx

(uˆ V 2 / 2)U w dxdy Fig. A.2

Mass flow through the element transports kinetic and thermal energy. Fig. A.2 shows energy convected in the x and y-directions only. Not shown is energy carried in the z-direction. To understand the components of energy transport shown in Fig. A.2, we examine the rate of energy entering the dydz surface. Mass flow rate through this area is U udydz . When this is 2 multiplied by internal and kinetic energy per unit mass, (u V / 2) , gives 2 the rate of energy entering dydz due to mass flow (u V / 2)U udydz. Similar expressions are obtained for the energy transported through all sides. Using the components shown in Fig. A.2 and including energy transfer in the z-direction (not shown) we obtain

B = (uˆ V 2 / 2) U u dydz (uˆ V 2 / 2) v dxdz (uˆ V 2 / 2) U wdxdy w (uˆ V 2 / 2) U u dydz >@(uˆ V 2 / 2) U u dxdydz wx w (uˆ V 2 / 2) U v dxdz >@(uˆ V 2 / 2) U v dxdydz wy w (uˆ V 2 / 2) U wdxdy >@(uˆ V 2 / 2) U w dxdydz . wz Simplifying

w 2 w 2 w 2 ½ B ® >@(uˆ V / 2) U u >@(uˆ V / 2) U v >@(uˆ V / 2) U w ¾u dxdydz. ¯wx w y w z ¿ APPENDIX A. Conservation of Energy: The Energy Equation 509

Making use of the definition of divergence (1.19) the above becomes

2 B ^ x >(uˆ V / 2) UV @ `dxdydz . (A-2)

(3) C = Net rate of heat addition by conduction

Let wqcc (qcc y dy)dxdz y w y qcc heat flux = rate of heat conduc- wqxcc tion per unit area qccdydz (qcxc dx)dydz x dy w x Fig. A.3 shows the z-plane of the dx element dxdydz. Taking into consid- qccdxdz eration conduction in the z-direction, y the net energy conducted through the Fig. A.3 element is given by

w qcxc C qcxcdydz qcycdxdz qczcdxdy (qcxc dx)dydz w x

w qcyc w qczc (qcyc dy)dxdz (qczc dz)dxdy. w y w z Simplifying

ªwqcxc wqcyc wqczc º C « »dxdydz . ¬ wx wy wz ¼ Introducing the definition of divergence

C ( x qcc) dxdydz . (A-3)

(4) D = Net rate of work done by the element on the surroundings

Rate of work is defined as forceu velocity. Thus

Rate of work = force u velocity.

Work done by the element on the surroundings is negative because it represents energy loss. We thus examine all forces acting on the element and their corresponding velocities. As we have done previously in the formulation of the equations of motion, we consider body and surface forces. Thus 510 APPENDIX A. Conservation of Energy: The Energy Equation

D Db Ds , (A-4) where

Db = Net rate of work done by body forces on the surroundings Ds = Net rate of work done by surface forces on the surroundings

ConsiderDb first. Let g x , g y and g z be the three components of gravitational acceleration. Thus Db is given by & V Db = U(g xu g y v g z w) dxdydz , or

D = (V x g) (A-5) & b U U g

To further clarify the negative sign consider a fluid particle Fig. A.4 being raised vertically, as shown Fig. A.4. If the particle is & & being raised, work is being done on it, and yet the dot product V g is negative; hence the negative sign being added to the work term to make the work positive.

Next we formulate an equation for rate of work done by surface stresses Ds . Fig. A.5 shows an element with some of the surface stresses. For the purpose of clarity, only stresses on two faces are shown. Each stress is associated with a velocity component. The product of stress, surface area and velocity represents rate of work done. Summing all such products, we obtain

wW W xz dx xz wx u W xy wu u dx V xx wx

W xz dz dy dx y z wW xy W xy dx x wx Fig. A.5 APPENDIX A. Conservation of Energy: The Energy Equation 511

§ wu ·§ wV xx · Ds ¨u dx¸¨V xx dx¸dydz u V xx dydz © wx ¹© wx ¹

§ ww ·§ wĲ xz · ¨w dx¸¨Ĳ xz dx¸dydz w Ĳ xz dydz © wx ¹© wx ¹ § wv ·§ wW xy · dx ¨ dx¸dydz dydz ¨v ¸¨W xy ¸ v W xy © wx ¹© wx ¹

§ ww ·§ wV zz · ¨w dz¸¨V zz dz¸dxdy w V zz dxdy © wz ¹© wz ¹ § wv ·§ wW zy · dz ¨ dz¸dxdy dxdy ¨v ¸¨W zy ¸ v W zy © wz ¹© wz ¹

§ wv ·§ wV yy · dy ¨ dy¸dxdz dxdz ¨v ¸¨V yy ¸ v V yy © wy ¹© wy ¹

§ wu ·§ wĲ yx · u dy ¨ dy¸dxdz u dxdz ¨ ¸¨Ĳ yx ¸ Ĳ yx © wy ¹© wy ¹

§ ww ·§ wĲ yz · w dy ¨ dy¸dxdz w dxdz ¨ ¸¨ Ĳ yz ¸ Ĳ yz © wy ¹© wy ¹

§ wu ·§ wĲ zx · ¨u dz¸¨Ĳ zx dz¸dxdy u Ĳ zx dxdy . © wz ¹© wz ¹

Note that negative sign indicates work is done by element on the surroundings. Neglecting higher order terms the above simplifies to

° § wV wW yx wW · § wW xy wV yy wW zy · D u¨ xx zx ¸ ¨ ¸ s ® ¨ ¸ v¨ ¸ ¯° © wx wy wz ¹ © wx wy wz ¹

§ wW wW yy wV · § wu wu wu · w¨ xz zz ¸ ¨ ¸ ¨V xx W yx W zx ¸ © wx wy wz ¹ © wx wy wz ¹ § wv wv wv · § ww ww ww ·½ ¨W xy V yy W zy ¸ ¨W xz W yz V zz ¸¾dxdydz. © wx wy wz ¹ © wx wy wz ¹¿ (A 6) Substituting (A-5) and (A-6) into (A-4) 512 APPENDIX A. Conservation of Energy: The Energy Equation

& & ª w D U V g dxdydz « (uV xx vW xy wW xz ) ¬wx w w º (uW yx vV yy wW yz ) (uW zx vW zy wV zz )»dxdydz. (A - 7) wy wz ¼

Substituting (A-1), (A-2), (A-3) and (A-7) into (2.14)

w ª § 1 2 ·º ª§ 1 2 · &º & & «U¨uˆ V ¸» «¨uˆ V ¸UV » qcc U V g wt ¬ © 2 ¹¼ ¬© 2 ¹ ¼ w w (uV xx vW xy wW xz ) (uW yx vV yy wW yz ) wx wy w (uW zx vW zy wV zz ). (A - 8) wz Note that equation (A-8) contains the nine normal and shearing stresses that appear in the formulation of the momentum equations (2.6). We will now use (2.6) to simplify (A-8). Multiplying equations (2.6a), (2.6b) and (2.6c) by the velocity components u, v and w, respectively, and adding the resulting three equations, we obtain

§ Du Dv Dw · U¨u v w ¸ U ug x vg y wg z © Dt Dt Dt ¹ § wV wW yx wW · § wW xy wV yy wW zy · § wW wW yz wV · u¨ xx zx ¸ ¨ ¸ w¨ xz zz ¸. ¨ ¸ v¨ ¸ ¨ ¸ © wx wy wz ¹ © wx wy wz ¹ © wx wy wz ¹ (A-9) However, 2 § Du Dv Dw· U DV U¨u v w ¸ , (A-10) © Dt Dt Dt ¹ 2 Dt and & & ug x vg y wg z V g (A-11)

Substituting (A-10) and (A-11) into (A-9) APPENDIX A. Conservation of Energy: The Energy Equation 513

* 2 U DV & & § wV wW yx wW · § wW xy wV yy wW zy · V g u¨ xx zx ¸ ¨ ¸ U ¨ ¸ v¨ ¸ 2 Dt © wx wy wz ¹ © wx wy wz ¹

§ wW wW yz wV · w¨ xz zz ¸ ¨ ¸ © wx wy wz ¹ . (A-12)

Returning to (A-8), the first and second terms are rewritten as follows

w ª § 1 2 ·º § 1 2 · wU w § 1 2 · «Uüˆ V ¸» üˆ V ¸ U üˆ V ¸ , (A-13) wt ¬ © 2 ¹¼ © 2 ¹ wt wt © 2 ¹

ª§ 1 2 · &º § 1 2 · & & § 1 2 · «üˆ V ¸UV » üˆ V ¸ UV UV üˆ V ¸ . (A-14) ¬© 2 ¹ ¼ © 2 ¹ © 2 ¹

Substituting (A-12), (A-13) and (A-14) into (A-8)

D § 1 2 · üˆ V ¸ Dt 2 0 © ¹ § 1 2 ·§ wU &· ª w § 1 2 · & § 1 2 ·º üˆ V ¸¨ UV ¸ U« üˆ V ¸ V üˆ V ¸» qcc © 2 ¹© wt ¹ ¬wt © 2 ¹ © 2 ¹¼ U DV 2 § wu wu wu · § wv wv wv · ¨V xx W yx W zx ¸ ¨W xy V yy W yz ¸ 2 Dt © wx wy wz ¹ © wx wy wz ¹ § ww ww ww · ¨W xz W yz V zz ¸ 0. © wx wy wz ¹ The above equation simplifies to

Duˆ § wu wu wu · § wv wv wv · U qcc ¨V xx W yx W zy ¸ ¨W xy V yy W zy ¸ Dt © wx wy wz ¹ © wx wy wz ¹ § ww ww ww · ¨W xz W yz V zz ¸. (A 15) © wx wy wz ¹

Equation (A-15) is based on the principle of conservation of energy. In addition, conservation of mass and momentum were used. Note that the only assumptions made so far are: continuum and negligible nuclear, electromagnetic and radiation energy transfer. We next introduce 514 APPENDIX A. Conservation of Energy: The Energy Equation constitutive equations to express the heat flux qcc in terms of the temperature field, and the normal and shearing stresses in terms of the velocity field. For the former we use Fourier’s law (1.8) and for the latter we apply Newtonian approximation (2.7). Application of Fourier’s law (1.8) gives the heat flux in the n-direction as wT qncc kn , (A-16) wn wherekn is thermal conductivity in the n-direction. Assuming isotropic material, we write kn k x k y k z k . (A-17) Using the operator , equation (A-16) is expressed as

qcc kT . (A-18) Substituting (A-18) and (2.7) into (A-15) and rearranging, we obtain Duˆ & U kT p V P) , (A-18) Dt where uˆ internal energy and ) is the dissipation function defined as

2 2 2 2 2 ª§ wu · § wv · § ww · º ªwv wu º ªwu wwº ) 2«¨ ¸ ¨ ¸ ¨ ¸ » wx ¨ wy ¸ wz «wx wy » « wz dx » ¬«© ¹ © ¹ © ¹ ¼» ¬ ¼ ¬ ¼ 2 2 ªww wvº 2 ªwu wv wwº « » « » . (A 19) ¬ wy wz ¼ 3 ¬wx wy wz ¼

Equation (A-18) is based on the following assumptions: (1) continuum, (2) negligible nuclear, electromagnetic and radiation energy, (3) isotropic material, and (4) Newtonian fluid. The next step is to express (A-18) first in terms of enthalpy and then in terms of temperature. Starting with the definition of enthalpy hˆ P hˆ uˆ . (A-20) U Differentiating (A-20) APPENDIX A. Conservation of Energy: The Energy Equation 515

Dhˆ Duˆ 1 Dp P DU . (A-21) Dt Dt U Dt U 2 Dt Substituting (A-21) into (A-18) 0 Dhˆ Dp p § DU *· U kT P) ¨ U V ¸ . (A-22) Dt Dt U © Dt ¹ Application of the continuity equation (2.2c) to (A-22) eliminates the last two terms. Thus (A-22) simplifies to Dhˆ Dp U kT P) . (A-23) Dt Dt We next express enthalpy in (A-23) in terms of temperature using the following thermodynamic relation [2]

ˆ 1 dh c p dT 1 ET dp , (A-24) U where E is the coefficient of thermal expansion, defined as

1 § wU · E ¨ ¸ . (A-25) U © wT ¹ p Taking the total derivative of (A-24) Dhˆ DT 1 Dp c p 1 ET . (A-26) Dt Dt U Dt Substituting (A-26) into (A-23) DT Dp Uc kT ET P) . (2.15) p Dt Dt

REFERENCES

[1] Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, 1960. [2] Van Wylen, G. J. and R.E. Sonntag, Fundamentals of Classical Thermodynamics, 2ndd ed., John Wiley & Sons, 1973. 516 APPENDIX B: Pohlhausen’s Solution

APPENDIX B: POHLHAUSEN’S SOLUTION

The transformed energy equation is

d 2T Pr dT f (K) 0 . (4.61) dK 2 2 dK The boundary conditions are T (0) 0 , (4.62a) T (f) 1, (4.62b) T (f) 1. (4.62c) Note that boundary conditions (4.60b) and (4.60c) coalesce into a single condition, as shown in (4.62b) and (4.62c). Equation (4.61) is solved by first separating the variables as dT d ( ) dK Pr f (K)dK . dT 2 dK Integrating the above from K 0 to K dT K d ( ) K dK Pr f (K)dK . dT 2 ³0 ³0 dK Evaluating the integral on the left-hand-side

dT K dK Pr ln f (K)dK . dT (0) 2 ³0 dK Taking the anti log of the above APPENDIX B. Pohlhausen’s Solution 517

K dT dT (0) ª Pr º exp« f (K)dK» . dK dK « 2 » ¬ ³0 ¼ Integrating again from K to K f and using boundary condition (4.62b) K f dT(0) f ª Pr º dT exp« f (K)dK» dK . ³K dK ³³K ¬ 2 0 ¼ This gives f K dT (0) ª Pr º T (K) 1 exp« f (K)dK» dK . (a) dK « 2 » ³³K ¬ 0 ¼ The constant dT (0) / dK in (a) is unknown. It is determined by satisfying boundary condition (4.62a), which gives

1 f K ½ dT (0) ° ª Pr º ° ® exp« f (K)dK» dK¾ . (b) dK « 2 » ¯°³³0 ¬ 0 ¼ ¿° Substituting (b) into (a) f K ª Pr º exp« f (K)dK» dK « 2 » ³³K ¬ 0 ¼ T (K) 1 . (c) f K ª Pr º exp« f (K)dK» dK « 2 » ³³0 ¬ 0 ¼

The integral in (c) can be simplified using the transformed momentum equation (4.44) d 3 f d 2 f 2 f (K) 0 . (4.44) dK 3 dK 2

Solving (4.44) for f (K) and integrating 518 APPENDIX B. Pohlhausen’s Solution

d 3 f d 2 f K K 1 dK 3 dK 2 f (K) dK dK ln . 2 d 2 f d 2 f (0) ³0 ³0 dK 2 dK 2

Multiplying both sides of the above by Pr and taking the anti log of the above Pr ªd 2 f º K « 2 » ª Pr º « dK » exp« f (K)dK» ¬ ¼ . (d) Pr « 2 » 2 ¬ ³0 ¼ ªd f (0)º « 2 » ¬« dK ¼» Substituting (d) into (c) gives f Pr ª 2 º d f dK « 2 » ³K ¬« dK ¼» T (K) 1 . (e) f Pr ª 2 º d f dK « 2 » ³0 ¬« dK ¼»

Similarly, substituting (d) into (b) gives the temperature gradient at the wall

Pr ªd 2 f (0)º « 2 » dT (0) « dK » ¬ ¼ . (4.63) dK f Pr ª 2 º d f dK ³0 « 2 » ¬« dK ¼» d 2 f (0) The constant in (f) is obtained from Table 4.1 dK 2 d 2 f (0) 0.332 . dK 2 APPENDIX B. Pohlhausen’s Solution 519

Thus (4.63) becomes

Pr dT (0) >@0.332 . (4.64) dK f Pr ª 2 º d f dK « 2 » ³0 ¬« dK ¼» 520 APPENDIX C: Laminar Boundary Layer Flow over Semi-Infinite Plate: Variable Surface Temperature

APPENDIX C

LAMINAR BOUNDARY LAYER FLOW OVER SEMI-INFINITE PLATE: VARIABLE SURFACE TEMPERATURE [1]

Surface temperature varies with distance along the plate according to

n Ts (x) Tf Cx . (4.72) Based on the assumptions listed in Section 4.3, temperature distribution is governed by energy equation (4.18)

wT wT w 2T u v D . (4.18) wx wy wy 2 The velocity components u and v in (4.18) are given in Blasius solution u df , (4.42) Vf dK

v 1 Q § df · ¨K f ¸ . (4.43) Vf 2 Vf x © dK ¹ The boundary conditions are:

n T(x,0) Ts Tf Cx , (4.74a)

T(x,f) Tf , (4.74b)

T(0, y) Tf . (4.74c)

The solution to (4.18) is obtained by the method of similarity transformation. We define a dimensionless temperature T as T T T s . (a) Tf Ts We assume that T (x, y) T (K) , (b) where APPENDIX C: Laminar Boundary Layer Flow over Semi-Infinite Plate: Variable Surface Temperature 521

V K y f . (c) Q x Equation (4.18) is transformed in terms of T (K) and K. Equation (a) is solved for T(x,y)

T Ts (Tf Ts )T . Substituting (4.72) in the above

n n T Tf Cx Cx T . (d)

The derivatives in (4.18) are formulated using (b)-(c) and the chain rule:

wT wT Cnxn1 Cnxn1T Cx n . wx wx However, wT dT wK K dT . wx dK wx 2x dK Substituting into the above

wT C dT Cnxn1 Cnxn1T xn1K . (e) wx 2 dK Similarly wT dT wK V dT Cxn Cxn f , (f) wy dK wy Q x dK

w 2T V d 2T Cxn1 f . (g) wy 2 Q dK 2 Substituting (4.42), (4.43) and (e)-(f) into (4.18)

df ª n1 n1 C n1 dT º Vf «Cnx Cnx T x K » dK ¬ 2 dK ¼ V Q ª df º V dT V d 2T f K f Cx n f D f . « » 2 2 Vf x ¬ dK ¼ Q x dK Q x dK 522 APPENDIX C: Laminar Boundary Layer Flow over Semi-Infinite Plate: Variable Surface Temperature

This simplifies to

d 2T df Pr dT nPr (1T ) f (K) 0 , (4.75) dK 2 dK 2 dK where Q Pr . D

REFERENCE

[1] Oosthuizen, P.H. and D. Naylor, Introduction to Convection Heat Transfer Analysis, McGraw-Hill, 1999. APPENDIX D: The von Kármán Momentum and Heat Transfer Analogy 523

APPENDIX D

THE VON KÁRMÁN MOMENTUM AND HEAT TRANSFER ANALOGY

The von Kármán Analogy is an extension of the Reynolds and Prandtl- Taylor Analogies that include a buffer layer between the viscous sublayer and outer layer. The viscous sublayer is described similar to (8.91):

qocc Ts Tb1 Pr ub1 , (a) W o c p where the subscript b1defines the lower edge of the buffer layer (which is also the edge of the viscous sublayer). As in the Prandtl-Taylor Analogy, a value of u y | 5 is chosen to approximate the edge of the viscous sublayer, leading to equation (8.95),

u C f 1 5 . (8.95) Vf 2 Combining the above with (a) gives

qocc C f Ts Tb1 Pr (5)Vf . (D-1) W o c p 2 For the buffer layer, von Kármán approximated the shape of the velocity profile as u 5.0ln y 3.05 , (D-2) which extends from y | 5 to y | 30. This region is a transitional one, where the flow field is dominated by molecular diffusion at one end, and by turbulent diffusion at the other. Unfortunately, the terms (Q H M ) and (D H H ) cannot be assumed to be equal in the buffer region; their ratio will vary from Q /D (= Pr) to H M / H H (|1) . Strictly speaking, then, the momentum and heat transfer equations are not analogous in this region. 524 APPENDIX D: The von Kármán Momentum and Heat Transfer Analogy

This region requires a different approach from the others. First, we can develop an expression H M as follows. From the velocity profile (D-2), the partial derivative of u is wu 5 . (a) wy y We can substitute this into the transformed Couette flow assumption, equation (8.52), to obtain

§ y · H Q ¨ 1¸ . (b) M ¨ ¸ © 5 ¹

Noting that H M H H , the ratio (Q H M ) / (D H H ) can be expressed as (Q H ) y m . (D H h ) y 5 1/ Pr 1 Substituting the above into (8.86), we obtain qcc y 5 1/ Pr 1 wT / wy c p . (D-3) W y wu / wy We can separate the derivatives and integrate with respect to y+ if we make two more adjustments to the equation: the first is to substitute variables y+ for y,u for u in the derivatives in (D-3). The second is to assume that qcc/W is constant; a reasonable assumption, since we saw that the ratio is in fact constant in the other regions. Thus (D-3) becomes

qccu* y wu wT o . (c) W oc p y 5(1/ Pr1) wy wy

* Finally, substituting u Vf C f / 2 in (D.3), and integrating both sides between y+ = 5 to y+ = 30, we obtain

qoccVf C f / 2 Tb2 Tb1 5ln(5Pr 1) , (D-4) W o c p APPENDIX D: The von Kármán Momentum and Heat Transfer Analogy 525 where Tb1 and Tb2 are the temperatures at each end of the buffer layer. The outer layer is described similar to that of the Prandtl Analogy,

qocc Tb2 Tf (Vf ub2 ) . (D-5) W o c p

Now, to obtain an expression for ub2 , we can use the velocity profile assumed for the buffer layer, equation. (D-2). Assuming that the buffer layer ends at y | 30 , ub2 | 5ln(30) 3 , (d) or, from the definition of u .

ub2 | 5ln 30 3 Vf C f / 2 . (e) Substituting these expressions into (D-5) gives* ª º qocc C f Tb2 T V «1 (5ln 30 3) » . (D-6) f W c f 2 o p ¬« ¼» Finally, adding (D-1), (D-4), and (D-6), and rearranging, the analogy becomes

Nu x C f / 2 St x { , (D-7) Rex Pr C f § 3 · ª5Pr 1º½ 1 5 ®¨ Pr ¸ ln« »¾ 2 ¯© 5 ¹ ¬ 30 ¼¿ This result is not significantly different, numerically, from von Kármán’s result, which is

* von Kármán used a slightly different approximation for ub2 , approximating it as ub2 ub1 (ub2 ub1 ) . The velocity ub1 was evaluated using (8.93), while ub2 ub1 |5ln(30/ 5) from the buffer layer velocity profile. The approximation used here yields virtually the same result. 526 APPENDIX D: The von Kármán Momentum and Heat Transfer Analogy3

C f / 2 Stx . (8.97) C f ª5Pr 1º½ 1 5 ®(Pr 1) ln« »¾ 2 ¯ ¬ 6 ¼¿ This expression gives the local heat flux for a flat plate with unheated starting length of xo .

REFERENCES

[1] von Kármán, T., “The Analogy between Fluid Friction and Heat Transfer,” Trans. ASME, Vol. 61, 1938, pp. 705-710. [2] Kakaç, S. and Yener, Y., Convective Heat Transfer, 2nd Ed., CRC Press, Boca Raton, 1995. APPENDIX E: Turbulent Heat Transfer from Flat Plate with 527 Unheated Starting Length

APPENDIX E

TURBULENT HEAT TRANSFER FROM A FLAT PLATE WITH UNHEATED STARTING LENGTH [1,2]

Consider a flat plate in turbulent flow, as depicted in Fig. 8.19. The velocity and temperature profiles are estimated using the 1/7th power law, 1/ 7 u § y · ¨ ¸ (8.65) Vf © G ¹ 1/ 7 T Ts § y · and ¨ ¸ . (8.114) Tf Ts © G t ¹ For this development, we will assume D Q (Pr 1) and H H H M ().Prt 1 The energy integral equation is as follows for constant-property, incompressible flow over an impermeable flat plate:

Gt (x) wT (x,0) d D u(T T )dy , (5.7) wy dx ³ f 0 Substituting the 1/7th-law velocity and temperature profiles into right-hand side of (5.7), and evaluating the integral, we obtain

The problem with this į T expression is that we can’t use V įt the 1/7th power law temperature profile to evaluate the left-hand of (E-1). This is x T because the gradient becomes o s undefined (that is, goes to Fig. 8.19 infinity) at y = 0. Recall from 528 APPENDIX E: Turbulent Heat Transfer from Flat Plate with Unheated Starting Length

Chapter 8 that we faced the same problem evaluating the wall shear in the momentum integral method. As a result, we have to develop some other means of estimating the wall heat transfer, one that does not result in an unrealistic value for the heat flux at the wall. It turns out that if we look at the heat flux anywhere in the boundary layer, and not simply at the wall, we can find an approach to the problem that works. Consider the definition of the apparent heat flux, qcc(x, y) wT (D H H ) . (8.41) Uc p wy

Since we are assuming D Q and H H H M , we can write (8.41) as qcc(x, y) wT (Q H M ) . (a) Uc p wy If we substitute the definition of the apparent shear stress, equation (8.40), for the term (Q H M ) into the above, qcc(x, y) W (x, y) wT / wy . (E-2) Uc p U wu / wy If we then substitute the 1/7th power law velocity and temperature profiles into this expression and simplify, we obtain 1/ 7 qcc(x, y) W (x, y) § G · (Ts Tf ) ¨ ¸ . (E-3) Uc p U © G t ¹ Vf Notice that the variable y does not appear in the above expression – it cancels out. The result is fortunate, because with y missing from (E-3), the heat flux does not become undefined (or zero, which is equally unrealistic) as y o 0 . The irony should not go unnoticed: even though we used the 1/7th power laws for velocity and temperature in the development, which both become undefined at the wall, the resulting expression yields a heat flux that is finite at the wall. Of course, we still need an expression for the shear stress W (x, y) anywhere in the boundary layer, which we have not attempted to model APPENDIX E: Turbulent Heat Transfer from Flat Plate with 529 Unheated Starting Length before. To model the shear stress, we can invoke an integral momentum equation, like equation (5.5), G (x) G (x) wu(x,0) d d Q V udy u 2dy . (5.5) wy f dx ³ dx ³ 0 0 Notice that this equation is integrated from y = 0, and yields an expression for shear stress at the wall. The problem with this is that we want an expression for shear stress at any y location. A more general integral equation requires a new derivation.

(i) Integral Momentum Equation for General y Location The development parallels the original development in Section 5.6, and so many of the details are left as an exercise. We will limit the derivation to incompressible flow over an impermeable flat plate. First, conservation of mass is applied to the dm y=į element shown in Fig. E.1. The derivation is e identical to the original development, and dmx yields the following expression for the mx mx dx differential mass flow at the boundary layer dx edge, dx y ªG (x) º d dm dme « Uudy»dx U v(y)dx . (E-4) y dx « ³ » ¬ y ¼ Fig. E.1 Application of the momentum theorem in the x-direction to the element depicted in Fig. E.2 gives F M (out) M (in) . (b) ¦ x x x What is different about this derivation is that x-momentum enters through the bottom of the element by way of the y-component of velocity. Applying the forces and momentum fluxes of Fig. E.2 to the conservation equation (b) gives § dp · d p(G y) ¨ p ¸dG p(G y) >@p(G y) dx Wdx © 2 ¹ dx

§ dM x · ¨M x dx¸ M x Vfdme u(U vdx) . (c) © dx ¹ 530 APPENDIX E: Turbulent Heat Transfer from Flat Plate with Unheated Starting Length

The momentum flux is G (x) 2 M x ³ Uu dy . (d) y Substituting (d) and (E-4) into (c), neglecting higher-order terms, and simplifying,

GG()x) (x dp d d G W (x, y) Uu 2dy V Uudy U>@V u(y) v(y) dx dx ³³f dx f y y . (E-5)

§ dp · y=į ¨ p ¸dG Vfdme © 2 ¹ p( y) G dM p(G y) M M x dx d>@p(G y) x x dx dx dx dx y W (x, y)dx u(y) Uv(y)dx (a) forces (b) x-momentum Fig. E.2

We need to solve (E-5) for the shear stress, W (x, y ) . For a flat plate, dp/dx is zero, and the 1/7th power law gives us an estimate for the velocity profile, u(y) . However, we still need an expression for the y component velocity, v(y) . (ii) Evaluation on of y velocity component We can determine an expression for v(y) by employing the differential conservation of mass, which for incompressible, two-dimensional flow is APPENDIX E: Turbulent Heat Transfer from Flat Plate with 531 Unheated Starting Length

wu wv 0 . (E-6) wx wy Rearranging and integrating from 0 to y, y y wv wu dy dy , (e) ³ wy ³ wx 0 0 y wu or v(y) v(0) dy . (f) ³ wx 0 We can evaluate the integral on the right side by invoking Leibniz’s rule for differentiating under the integral sign: for an integrand f (x, y) with limits of integration a(x) andb(x) , b b d w db da f (x, y)dy f (x, y)dy f (b, x) f (a, x) . (E-7) dx ³ ³ wx dx dx a a For our purposes, f o u , ayo 0 , and b o . Since both limits of integration are not functions of x, then (e) becomes simply y d v(y) udy . (g) dx ³ 0 Substituting the 1/7th power law (8.65), into (f), and evaluating, it can be shown that 8/ 7 V § y · dG v(y) f ¨ ¸ . (E-8) 8 © G ¹ dx (iii) Evaluation of Shear Stress W (x, y) Expressing equation (E-5) in terms of the shear stress, GG()x) (x d d W (x, y) Uu 2dy V Uudy U>@V u(y) v(y) (E-9) dx ³³f dx f y y . 532 APPENDIX E: Turbulent Heat Transfer from Flat Plate with Unheated Starting Length

Substituting the velocity expressions (8.65) and (E-9), it can be shown that the shear stress can be expressed as

9/ 7 W (x, y) 7 2 ª § y · º dG Vf «1 ¨ ¸ » . (E-10) U 72 ¬« © G ¹ ¼» dx Note that at y = 0, we find that the wall stress is W 7 dG o V 2 , U 72 f dx Which is exactly what we found in the Prandtl-von Karman integral solution (8.68). In fact, we can express (E-10) as the following ratio:

9/ 7 W (x, y) § y · 1 ¨ ¸ . (E-11) W o © G ¹ Equation (E-11) is more convenient than (E-10) in that we have already solved for the wall shear stress in the Prandtl-von Karman integral solution. This will become convenient as we move forward with the derivation.

(iii) Evaluation of Heat Flux qcc(x, y) With an expression for the shear stress W (x, y ) in hand, we can substitute it into (E-3) to obtain an expression for the heat flux qcc(x, y) . The result is

1/ 7 9/ 7 qcc(x, y) § G · (T T ) W ª § y · º ¨ ¸ s f o «1 ¨ ¸ » , (h) c ¨ ¸ V U p © G t ¹ f U ¬« © G ¹ ¼» or, from the definition of friction factor,

1/ 7 9/ 7 qcc(x, y) § G · (T T ) § 2 C f ·ª § y · º ¨ ¸ s f ¨V ¸«1 ¨ ¸ » . (i) c ¨ ¸ V ¨ f 2 ¸ U p © G t ¹ f © ¹¬« © G ¹ ¼» Finally, we can evaluate this expression at the wall (y = 0), which gives

1/ 7 qcc § G · C f o ¨ ¸ (T T )V . (E-12) ¨ ¸ s f f Uc p © Gt ¹ 2 (iv) Solution of Integral Energy Equation APPENDIX E: Turbulent Heat Transfer from Flat Plate with 533 Unheated Starting Length

Finally, we can substitute our expression for the heat flux (E-12) into the integral energy equation, (E-1). Doing so, and simplifying, we obtain

1/7 8/ 7 § G · § C f · d § 7 G · ¨ ¸ ¨ ¸ ¨ t ¸ . (E-13) ¨ ¸ ¨ ¸ ¨ 1/ 7 ¸ © Gt ¹ © 2 ¹ dx © 72 G ¹ To evaluate this expression, we note that the Prandtl-von Karman integral solution gives us a solution for the friction factor,

C f 0.02968 1/ 5 . (8.71) 2 Rex Substituting this into (E-13) gives

We will find it easier to evaluate (E-14) if we define [ Gt /G. Equation (E-14) can be written as

1/5 7 d 8/ 7 1/ 7 §V x · G[ 0.02968 [ ¨ f ¸ . (j) 72 dx © Q ¹ Applying the product rule to the derivative, (i) can be written as

1/ 5 8 1/ 7 d[ 8/ 7 dG § 72 · 1/ 7 §V x · G [ [ 0.02968¨ ¸[ ¨ f ¸ , 7 dx dx © 7 ¹ © Q ¹ or, multiplying both sides by [ 1/ 7 ,

1/5 8 2/7 d[ 9/ 7 dG § 72 ·§V x · G [ [ 0.02968¨ ¸¨ f ¸ . (k) 7 dx dx © 7 ¹© Q ¹ We can again invoke the Prandtl-von Karman solution, and substitute (8.67) for the velocity boundary layer: G 0.3816 1/ 5 . (8.70) x Rex Doing this, we obtain 534 APPENDIX E: Turbulent Heat Transfer from Flat Plate with Unheated Starting Length

8 2/ 7 d[ 9/7 4 1 0.02968 § 72 · 1 [ [ x ¨ ¸x . (l) 7 dx 5 0.3816 © 7 ¹ Now, by the product rule, d[ 7 d[ 9/ 7 [ 2/ 7 , dx 9 dx So (l) becomes

9/ 7 d[ 9 9/ 7 1 0.02968 § 72 ·§ 9 · 1 [ x ¨ ¸¨ ¸x . (E-15) dx 10 0.3816 © 7 ¹© 8 ¹ To solve this equation, we first note that (E-15) is of the form dY AY B , dx x x whereY [ 9 / 7 and A B 0.9 . This type of equation can be solved by use of an integrating factor, the result being

9/ 7 § G t · 10 C ¨ ¸ B 9/10 , (n) © G ¹ 9 x where C is a constant of integration. To evaluate this constant, we can invoke the boundary condition G t = 0 at x = xo . This gives 10 C Bx9 /10 , (o) 9 o and so (n) becomes APPENDIX E: Turbulent Heat Transfer from Flat Plate with 535 Unheated Starting Length

9 /10 7 / 9 G ª § x · º t «1 ¨ o ¸ » . (E-16) G x ¬« © ¹ ¼» Finally, we can use this in our expression for the heat flux (E-12), 1/ 9 ª 9 /10 º qocc C f § xo · (Ts T )V «1 ¨ ¸ » , (p) Uc f f 2 x p ¬« © ¹ ¼» And rearranging, we can show that

1/ 9 ª 9 /10 º qocc C f § xo · { St x «1 ¨ ¸ » (8.121) Uc V (T T ) 2 x p f s f ¬« © ¹ ¼» This expression gives the local heat flux for a flat plate with unheated starting length of xo .

REFERENCES

[1] Burmeister, L.C., Convective Heat Transfer, 2nd Ed., John Wiley and Sons, New York, 1993. [2] Kays, W.M., Crawford, M.E., and Weigand, B., Convective Heat and Mass Transfer, 4th Ed., McGraw-Hill, Boston, 2005. 536 APPENDIX F: Properties of Air at Atmospheric Pressure

APPENDIX F: Properties of Air at Atmospheric Pressure

T Cp U P Q k Pr oC J/kg-oC kg/m3 kg/s-m m2/s W/m-oC

40 1006.0 1.5141 15.17u106 10.02u106 0.02086 0.731 30 1005.8 1.4518 15.69u106 10.81u106 0.02168 0.728 20 1005.7 1.3944 16.20u106 11.62u106 0.02249 0.724 10 1005.6 1.3414 16.71u106 12.46u106 0.02329 0.721 0 1005.7 1.2923 17.20u106 13.31u106 0.02408 0.718 10 1005.8 1.2467 17.69u106 14.19u106 0.02487 0.716 20 1006.1 1.2042 18.17u106 15.09u106 0.02564 0.713 30 1006.4 1.1644 18.65u106 16.01u106 0.02638 0.712 40 1006.8 1.1273 19.11u106 16.96u106 0.02710 0.710 50 1007.4 1.0924 19.57u106 17.92u106 0.02781 0.709 60 1008.0 1.0596 20.03u106 18.90u106 0.02852 0.708 70 1008.7 1.0287 20.47u106 19.90u106 0.02922 0.707 80 1009.5 0.9996 20.92u106 20.92u106 0.02991 0.706 90 1010.3 0.9721 21.35u106 21.96u106 0.03059 0.705 100 1011.3 0.9460 21.78u106 23.02u106 0.03127 0.704 110 1012.3 0.9213 22.20u106 24.10u106 0.03194 0.704 120 1013.4 0.8979 22.62u106 25.19u106 0.03261 0.703 130 1014.6 0.8756 23.03u106 26.31u106 0.03328 0.702 140 1015.9 0.8544 23.44u106 27.44u106 0.03394 0.702 150 1017.2 0.8342 23.84u106 28.58u106 0.03459 0.701 160 1018.6 0.8150 24.24u106 29.75u106 0.03525 0.701 170 1020.1 0.7966 24.63u106 30.93u106 0.03589 0.700 180 1021.7 0.7790 25.03u106 32.13u106 0.03654 0.700 190 1023.3 0.7622 25.41u106 33.34u106 0.03718 0.699 200 1025.0 0.7461 25.79u106 34.57u106 0.03781 0.699 210 1026.8 0.7306 26.17u106 35.82u106 0.03845 0.699 220 1028.6 0.7158 26.54u106 37.08u106 0.03908 0.699 230 1030.5 0.7016 26.91u106 38.36u106 0.03971 0.698 240 1032.4 0.6879 27.27u106 39.65u106 0.04033 0.698 250 1034.4 0.6748 27.64u106 40.96u106 0.04095 0.698 260 1036.5 0.6621 27.99u106 42.28u106 0.04157 0.698 270 1038.6 0.6499 28.35u106 43.62u106 0.04218 0.698 280 1040.7 0.6382 28.70u106 44.97u106 0.04279 0.698 290 1042.9 0.6268 29.05u106 46.34u106 0.04340 0.698 300 1045.2 0.6159 29.39u106 47.72u106 0.04401 0.698 310 1047.5 0.6053 29.73u106 49.12u106 0.04461 0.698 320 1049.9 0.5951 30.07u106 50.53u106 0.04521 0.698 330 1052.3 0.5853 30.41u106 51.95u106 0.04584 0.698 340 1054.4 0.5757 30.74u106 53.39u106 0.04638 0.699 350 1056.8 0.5665 31.07u106 54.85u106 0.04692 0.700 APPENDIX G: Properties of Saturated Water 537

APPENDIX G: Properties of Saturated Water Index 539

Index

A absorptivity, 8 closure problem of turbulence, 315 accuracy coefficient of thermal expansion, correlation equations, 389 39, 46, 47, 260 integral method, 163 Colebrook, C.F., 372, 380, 381 algebraic method, 349, 377 Colburn analogy, 342, 345 turbulent flat plate flow, 349 Cole’s law of the wake, 328 turbulent pipe flow, 336 compressibility, 39, 443, 447, 450, analogies for momentum and 462, 467-470, 477, 485- heat transfer, 336 487, 493, 495 Colburn, 342, 343, 345 conduction sublayer, 345, 347, Prandtl-Taylor, 359 348, 351, 358, 359 Reynolds, for flat plate, 337 conservation of for pipe flow, 374 energy, 37, 38, 507-515 validity of, 344 mass, 22 von Kármán momentum, 27 for flat flow, 342 constitutive equations, 29 for pipe flow, 523, 526 continuity equation, 22 apparent heat flux, 318 Cartesian coordinates, 22 apparent shear stress, 315, 318, cylindrical coordinates, 24 340, 341, 346 spherical coordinates, 25 axial conduction, 109, 113, 213 continuous law of the wall, 326 223, 228, 233, 236, 242, continuum, 2, 437-445 262, 439, 440, 442, 447, convective derivative, 11 470, 477, 480-483, 485, correlation equations, 387-390, 486, 494, 496-498 accuracy, 389 B enclosures, 413-418 Blasius, 118, 119-131, 140, 144 external forced convection, body force, 28, 39 388, 390-393 boundary conditions, 48, 204, cylinders, 396 264, plates, 390-393 temperature jump, 448-449 spheres, 397 velocity slip, 448-449 external free convection, 405- boundary layer, 99 410 thermal, 100 horizontal cylinders, 409-410 thickness, 103, 108, 110, 116, horizontal plates, 408-409 117, 121, 123, 125, 128 inclined plates, 407-408 viscous, 100, 101, 118, 269 spheres, 410 Boussinesq vertical cylinders, 409 approximation, 46, 261 vertical plates, 405-407 hypothesis in turbulent flow, internal forced convection, 397- 314, 317, 318 398 C entrance region, 397-398 Cartesian coordinates, 22, 27, 37 540 Index

387, 396-397 fully develop, 403, 404 F limitations, 389 film temperature, 130, 271, 275, non-circular channels, 404-405 344, 389-390, 393, 409 other correlations, 422 flow classification procedure for selecting, 389- continuum, no-slip, 441-442 390 continuum, slip, 441-442 Couette flow, 71, 322-325, 345, free molecular, 441-442 366, 439, 345, 449-451, transition, 441-442 459-460, 462 forced convection, 21, 54 Cylindrical coordinates, 24, 32, correlation, 387-404 41, 42, 486 Fourier’s law, 3 D free convection, 21, 259 differential formulation, 21, 161 correlations, 405-410, 413-418 dimensionless variables, 52 enclosures, 413-418 Direct Numerical Simulation, 303 inclined plates, 407 dissipation, 39-43, 53, 439, 442, free molecular flow, 441 443, 447-448, 450, 456 friction coefficient, 117, 122, 172, Dittus-Boelter correlation, 376 444 divergence, 9 friction factor, 404, 444 E friction velocity, 321, 350, 358 Eckert number, 53, 447 fully developed flow, 70, 205, 214, eddies, 298 236, 238, 243, 444, 445, eddy 493 conductivity, 317 fully developed temperature, 206, diffusivity, 317, 318, 319, 323, 229-237, 471, 482, 488, 325, 326 489 turnover time, 300 fully turbulent region (law of the viscosity, 317, 319 wall), 324-328, 332-334, 348, emissivity, 8 350,354, 359 energy cascade, 300 G energy equation, 37-48, 53, 69, Gnielinski, 376, 379, 382, 404 101, 109-111, 123, 125, governing equations, 51, 54, 260, 141, 146, 165, 175, 179, 443 180, 183, 186, 190, 228, gradient, 10 233, 236, 243, 252, 261, Graetz, 242, 496 263, 442, 447, 446, 455, Grashof number, 53, 260, 264 470, 480, 488, 496, 507 H entrance length, 205-209 heat flux, 4, 123, 204, 212, 225, entrance region, 203-207, 225 231, 236, 237 entry length, turbulent flow, 361, heat transfer coefficient, 6, 55, 86, 362 123, 128, 136, 148, 177, exact solution, 69, 71, 161, 185, 180, 190, 222, 226, 245, 288 267, 273, 285, 290, 391, external flow, 99, 115, 164, 166, 400, 440-441, 450, 454- 167, 206, 206, 207, 225, 455, 457, 469 Index 541

homogeneous turbulence, 303 mass flow rate, 451, 453, 455, horseshoe vortex, 298 462, 468, 469, 480, 483, hydrodynamic entrance length, 487 206-209 mean free path, 2, 437-443, 463 hydrodynamic entrance region, mean temperature, 211-220, 233, 206, 209 235, 244-246, 269, 398, I 400, 454, 455, 469, 470, ideal gas, 41,460, 463, 465, 468, 473, 475, 481, 483, 488, 477, 484, 494, 500, 501 491, 496 inclined plates, 279, 407 microchannels, 437, 440-443, integral method, 161, 279 445-447, 464, 483 for heat transfer coefficient, micron, 440 351, 526 mixing, laminar turbulent, 294 for turbulent flow, 351, 326 Momentum integral, turbulent formulation, 161, 279 flow, 328 momentum, 328-330 Moody chart, 372 solutions, 170, 283 N isotropic, 5 Navier-Stokes equations, 27, 31, isotropic turbulence, 303 32, 53, 99, 437, 442, 446, J 486 joule, 13 Newton, 13 K Newton’s law of cooling, 6 Kármán-Nikuradse, 371 Kestin, 333 Nikurdadse, 368-369, 371, 372, Knudsen number, 2, 437-443, 381 447, 453, 454, 458, 463, no-slip condition, 48, 100 464, 463, 466, 467, 476, no-slip flow, 462 477, 482, 485, 493, 498 no-temperature jump, 437, 441 Kolmogorov, 301, 353 no-velocity slip, 437 Kolmogorov microscale, 301-302 Nusselt number, 55, 129, 142, L 149, 177, 179, 186, 224, laminar boundary layer, 99, 116, 227, 230, 231, 235, 237, 140, 143, 520-522 245, 246, 253, 268, 285, laminar mixing294 287, 288, 389-392, 397, law of the wall, 324-328, 333- 398, 403, 405, 406, 409, 334, 346, 350 410, 414, 446, 454, 455- local derivative, 11 459, 469-476, 478, 487, local Nusselt number, 55, 124, 493, 498 129, 142, 149, 178, 179, O 184, 254, 268, 269, 275, operator , 9 276, 285, 343, 361, 389- other correlations, 328 392, 405 P M Peclet number, 100, 112, 228, 396, Mach number, 447, 448 447, 498

542 Index

Petukhov, 376, 379-380, 382 Richardson, 300, 354 Pohlhausen, 125, 127, 130, 392, rotating flow, 86 516-419 S Poiseuille, 77, 449-450, 462-463, scale analysis, 59, 103, 110, 117, 480, 485-486, 496 123, 206, 224, 231 Poiseuille number, 445 slip flow, 442, 444, 449-450, 459, Prandtl 460, 477, 484 law for smooth pipe, 369-371 similarity parameters, 54, 55 mixing-length theory, 318-320 similarity transformation, 119, number, 53, 54, 66, 68, 101, 125, 141, 146, 248, 274 126-129, 143, 147, 148, Spalding, 326, 327, 333, 354 175, 187, 192, 207, 212, spherical coordinates, 21, 25, 33, 225, 228, 236, 260, 266, 34, 42 267, 269, 275, 280, 288, surface roughness, 334, 335, 345, 289, 336, 338, 339, 342, 352, 371-372, 380 344, 345, 347-349, 377, Stanton number, 339, 350, 374, 380, 381, 388, 396, 403, 375, 379 414, 443 Stefan-Boltzmann law, 8 Prandtl-Taylor analogy, 339-342 substantial derivative, 11 Prandtl-von Kármán model for surface force, 28, 438-439 turbulent friction factor, T 328-330 Taylor, 319, 326, 339-341, 345, Properties 351, 354 air, 536 temperature jump, 438, 441-443, water, 537 448-449, 455, 458-459, R 476, 486, 488 rarefaction, 439, 443, 458, 462- thermal 463, 463, 464, 467, 469, boundary layer, 100, 109-111, 477, 485, 486, 487, 493, 113, 122, 123, 125, 128, 498 136, 153, 169, 174, , 178, Rayleigh number, 187, 188, 195, 183, 191, 260, 262, 272, 208, 225, 260, 261, 271, 279 275, 276, 278, 287 conductivity, 4 Reichardt, 327, 354 tables, 536, 537 Reynolds diffusivity, 260 analogy, 337, 374, 342, 345, entrance length, 206, 208, 209 351, 360 entrance region, 242 decomposition, 304-305, 309 thermodynamic equilibrium, 2, 3, heat flux, 315, 317 437-438, 441-444 number, 22, 53, 54, 100, 108, total derivative, 11 112, 122, 130, 149, 177, total differential, 10 184, 188, 206, 207, 226, transition 228, 240, 241, 296, 299- flow, 441 302, 335, 354-357, 390, Rayleigh number, 260 403, 445, 458, 484 Reynolds number, 22, 203 stress, 314, 315, 317, 324 Turbulence, Index 543

homogeneous, 303 turbulent pipe flow isotropic, 303 1/7th law onset of, 296 temperature profile, 376 scales, 299-302 velocity profile, 367-369 turbulent apparent heat flux, 363 boundary layer equations, 311- apparent shear stress, 363 313 effect of surface roughness, energy, 313-314 334-335, 344, 371-372 momentum, 311-313 effect of pressure gradient, flow, 22 327-328 heat flux, 315 entry length, 361-362 mixing, 294 mean temperature, 363-364 Prandtl number, 336, 338, 344, mean velocity, 363-364 347, 348, 355 universal temperature profile, shear stress, 315 377-379 temperature profile, 345-346 U velocity defect law, 295, 297, universal velocity profile, 364-367 320-322 units, 12 velocity profile, 320-327 V turbulent boundary Layer van Driest, 326, 354 1/7th law variable surface temperature, 140, temperature profile, 349 426 velocity profile, 329-330 velocity conduction sublayer, 347, 348 defect law, 320-322 Cole’s law of the wake, 328, derivative, 9 354 slip, 437-439, 441-442, 444, law of the wall (temperature), 448-449, 459 349 vector, 9 law of the wall (velocity), 324, viscous boundary layer, 70, 87 325, 331, 354 viscous sublayer, 323, 324 overlap region, 327 von Kármán analogy, 342, 345, universal temperature profile, 352, 375-376, 523-526 345-349 von Kármán constant, 325 universal velocity profile, 320- von Kármán, 326-329, 332-333, 328 355, 367-368, 371, 375, viscosity sublayer, 324 381 wake region, 327-328 vortex filament, 298 wall coordinates, 322 turbulent flow conservation W equations wall coordinates, 322 energy, 310 watt, 13 mass, 307-308 White, 327, 332-333, 362, 372, momentum, 308-309 380