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Theoret. Comput. Fluid Dynamics (2001) 14: 377–398Theoretical and ComputationalFluid Dynamics Springer-Verlag 2001

Breakdown of the Reynolds Analogy in a Stagnation RegionUnder Inflow Disturbances

Sungwon Bae and Hyung Jin Sung

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology,373-1 Kusong-dong, Yusong-ku, Taejon 305-701, Korea

Communicated by M.Y. Hussaini

Received 12 May 2000 and accepted 6 March 2001

Abstract. A systematic analysis is performed for the Reynolds analogy breakdown at stagnation-regionflow and heat transfer in the presence of inflow disturbances. The Reynolds analogy breakdown betweenmomentum and energy transfers in a stagnation region is scrutinized by varying the Reynolds number(5000 ≤ Re ≤ 20 000), the amplitude (0.00075 ≤ A ≤ 0.003) and the length scale (λ/δ = 10.6). A span-wise sinusoidal variation is superimposed on the velocity component normal to the wall. Self-similaritysolutions are obtained with trigonometric series expansions. The Reynolds analogy criterion demonstratesthat the rate of change of skin friction is different from that of wall heat transfer. Different evolutions ofthe rates of skin friction and wall heat transfer are due to the difference between 〈s′v′〉 and 〈v′T ′〉. Anin-depth analysis on 〈s′v′〉 and 〈v′T ′〉 is performed by analysis using disturbance correlations based onthe fluctuating velocity transport equations in vorticity form. It is found that the pressure fluctuations, thewall blocking and the Lamb vectors are responsible for the breakdown of the Reynolds analogy. A directcomparison is made between momentum and energy balances associated with the three responsible mech-anisms. A common finding is that their profiles are changed significantly at a location where the evolutionof the streamwise vortex is strong.

1. Introduction

In general, turbulent convective heat transfer is predicted by employing the turbulent Prandtl number (Prt), inwhich the eddy diffusivity for heat (αt) is prescribed through the known eddy diffusivity for momentum (νt).The assumption Prt (= αt/νt) = constant is the fundamental form of what is called the Reynolds analogy.A literature survey reveals that many studies on convective heat transfer have been successfully performed byadopting the Reynolds analogy (Cebeci, 1973; Reynolds, 1975; Gaviglio, 1987). However, a breakdown ofthe Reynolds analogy occurs in some cases. Huang et al. (1995) addressed the breakdown in a compressiblechannel flow. Wang and Campbell (1992) proposed two modes of momentum transfer, i.e., a slow streamingmode and a fast impulsive mode, whereas energy transfer has only the slow streaming mode. This may sug-gest that the mechanism of momentum transfer is generically different from that of heat transfer. It is knownthat the Reynolds analogy is broken down in complex and nonequilibrium flows (Simonich and Bradshaw,1978). Of particular interest is stagnation-region heat transfer, in which wall heat transfer is significantlyaugmented compared with skin friction. Furthermore, stagnation-region heat transfer in the presence of free-stream disturbances is of prime importance in many engineering applications. An in-depth analysis is neededto look into the breakdown in stagnation-region heat transfer.

This work was supported by a grant from the National Research Laboratory of the Ministry of Science and Technology, Korea.

377

378 S. Bae and H.J. Sung

Stagnation-region heat transfer augmentation in the presence of free-stream disturbances is caused by theamplification of streamwise vorticity (Van Fossen and Simoneau, 1987; Bae et al., 2000). The streamwisevorticity, which is imposed at the inflow boundary, is convected into the stagnation region, and is ampli-fied by stretching due to the mean strain rate (Sutera, 1965). Many experimental and numerical studies havebeen made to determine the influence of inflow disturbances on stagnation-region heat transfer (Ames andMoffat, 1990; Maciejewski and Moffat, 1992a,b; Van Fossen et al., 1995). Van Fossen et al. (1995) investi-gated the effects of free-stream parameters such as turbulence intensity, length scale and Reynolds numberon stagnation-region heat transfer in a turbine blade. Ames and Moffat (1990) made a mock combustor andsimulated experimentally the leading-edge flow and heat transfer. They found that the length scale of free-stream turbulence is inversely correlated with the heat transfer rate, which is related with the wall blockingeffect proposed by Hunt and Graham (1978).

Recently, Bae et al. (2000) performed numerical simulations of stagnation-region heat transfer in thepresence of inflow disturbances. The sensitivity of heat transfer to free-stream vorticity was scrutinized byvarying the length scale, intensity and Reynolds number. As an organized inflow disturbance, a spanwisesinusoidal variation was superimposed on the velocity component normal to the wall. The compressibleNavier–Stokes equations and energy equation were numerically solved. The computational results dis-closed the detailed behavior of streamwise vortices near the stagnation region. Three regimes of behaviorwere found depending on the length scale: “damping,” “attached amplifying” and “detached amplifying”regimes.

The aforementioned numerical simulation of Bae et al. (2000) provided a better understanding ofstagnation-region heat transfer augmentation in the presence of inflow disturbances. However, the detailedmechanism of the Reynolds analogy breakdown was not dealt with. To see the salient characteristics of theReynolds analogy breakdown in the vicinity of the wall for the present study, the governing equations areanalytically solved with trigonometric series expansions. Since energy is transferred from a large scale tosmall scales driven by vortex stretching, it is expected that the evolution of the small scales can be capturedwell with the trigonometric scales. The objective of this study is to examine the breakdown mechanism inthe presence of inflow disturbances. As an organized inflow disturbance, a simple sinusoidal variation issuperimposed on the velocity component normal to the wall. The influence of inflow disturbances on theamplification of vorticity and its heat transfer augmentation in the stagnation region is investigated based onthe analytical solutions. The present analytical solutions are compared with the results of the earlier work ofnumerical simulation (Bae et al., 2000).

The Reynolds number Re based on a reference length scale L is varied from 5000 to 20 000. Themagnitude A is changed from 0.00075 to 0.003. The length scale λ associated with the organized inflow dis-turbance is fixed at λ/δ = 10.6, where δ is a 99% undisturbed laminar boundary layer thickness. The Prandtlnumber is Pr = 1.0. The wall temperature is isothermal with Tw/T0 = 0.99, where Tw is the wall temperatureand T0 is the free-stream total temperature.

2. Mathematical Formulation

The physical situation of interest is that of a viscous, incompressible, steady flow in the neighborhood ofa stagnation region into which a certain amount of vorticity is transported by the main stream (Sutera, 1965).A schematic diagram of the flow configuration is shown in Figure 1. In the presence of inflow disturbances,we are particularly interested in observing the evolution of this vorticity as it penetrates the boundary layerand the contaminant effects produced on the average value of shear stress and heat transfer at the boundarylayer. A coordinate system with origin at the solid boundary is used, which coincides with the (x, z) plane.The mean flow approaches the wall from y = ∞, where it is parallel to the y axis. It divides into two equalsemi-infinite streams at the wall which flow away from the stagnation region along the positive and negativex axes. The z axis is perpendicular to the plane of flow.

The governing equations, incompressible Navier–Stokes, energy and continuity equations are madedimensionless using the following reference values: the reference quantities u∞, T∞ and L represent a vel-ocity, temperature and length scale, respectively. The Reynolds number is defined as Re = u∞L/µ∞. Here,

Breakdown of the Reynolds Analogy in a Stagnation Region Under Inflow Disturbances 379

λ

x

y

zwall

A

v(x,z)=vm(x)1+A cos(2 πz/λ)

ωx

Figure 1. Schematic diagram of the present study.

L is the distance from the wall to the inflow boundary. The continuity equation for an incompressible flow is

∂ui

∂xi= 0 , (1)

where xi (i = 1, 2, 3) denotes three spatial directions in which x is the direction parallel to the wall, y denotesthe direction normal to the wall and z is the spanwise direction, respectively.

The steady incompressible Navier–Stokes equations are

uj∂ui

∂xj= − ∂p

∂xi+ 1

Re

∂2ui

∂xj2 , (2)

where ui is the ith velocity component in the xi direction (i = 1, 2, 3) and p is the pressure.The steady energy equation with no dissipation is

uj∂T

∂xj= 1

RePr

∂2T

∂xj2 . (3)

In the above equation, Pr is the Prandtl number, Pr = µCp/k where µ is the viscosity, Cp is the specific heatat constant pressure and k is the thermal conductivity.

In a manner similar to the Hiemenz type, a self-similarity is assumed in the x direction (Sutera et al.,1963). We now investigate solutions of the type

u = x × s(y, z),

v = v(y, z),

w = w(y, z), (4)

p = pt − 1

2x2 + p(y, z),

T = T(y, z).

Such solutions differ from those of Sutera et al. (1963) in that they described a three-dimensional vortic-ity field without pressure. In (4) pt represents the total pressure and s(y, z) is a strain rate. All the variablesin (4) are independent of the x direction due to self-similarity.


At this point, the nature of periodic three-dimensionality is specified, which is capable of providingvorticity with the desired orientation. We seek solutions which are periodic in z.

u = x

sm(y)+

∞∑n=1

san(y) cos(kn z)+ sbn(y) sin(kn z)

,

v = vm(y)+∞∑

n=1

van(y) cos(kn z)+vbn(y) sin(kn z),

w =∞∑

n=1

wan(y) cos(kn z)+wbn(y) sin(kn z), (5)

p = pt − 1

2x2 + pm(y)+

∞∑n=1

pan(y) cos(kn z)+ pbn(y) sin(kn z),

T = Tm(y)+∞∑

n=1

Tan(y) cos(kn z)+ Tbn(y) sin(kn z),

where the subscript “m” denotes n = 0 and the quantity kn is the dimensionless wave number of the nthharmonic component of the periodically disturbed vorticity. kn is defined as kn = 2nπ/λ, where λ is thedimensionless fundamental wave length in the spanwise direction. The dimensionless wave length corres-ponding to the nth harmonic component of the vorticity is simply λn = 2π/kn . If we now substitute (5) intothe governing equations and collect terms in terms of sin(kn z) and cos(kn z), we obtain a system of ordinarydifferential equations in the y direction. Details regarding the ordinary differential equations can be found inAppendix.

As shown in Figure 1, a sinusoidal disturbance is imposed on the v velocity at the inflow boundary:

v(y, z) = vm(y)(1+ A cos(k1z)), (6)

where vm denotes the mean velocity averaged in the spanwise direction at the inflow boundary. A representsthe magnitude of the inflow disturbance.

The boundary conditions for the variables in (5) are:At the wall:

sm = vm = 0,

Tm = Tw,

san = sbn = van = vbn = wan = wbn = 0, (7)

Tan = Tbn = 0,

where Tw is the isothermal wall temperature, which is set to 0.99 for this study.At the inflow boundary:

sm = 1.0,

dvm

dy= −1.0,

va1 = A,

van = 0 for n ≥ 2,

san = sbn = vbn = 0, (8)


wan = sbn +vbn

kn,

wbn = −san +van

kn,

Tan = Tbn = 0,

dpan

dy= 0,

pbn = 0.

The equations which are ordinary, nonlinear and coupled are numerically solved by using the boundary valueproblem solver developed in the IMSL library.

3. Results and Discussion

Before proceeding further, it is important to ascertain the accuracy and reliability of the present analytic so-lution. Toward this end, a series solution has been obtained for an example case: λ/δ = 5.3, A = −0.007and Re = 10 000. The convergence of s(y, z) is shown in Figure 2, where a partial sum of N terms iss(y, z) = ∑N

n=1 san(y) cos(kn z). Figure 2 exhibits the convergence of s(y, 0) as the partial sum of N termsis increased. The inset shows that increasing the limit N of the sum gives better approximations to the solu-tion. When N is less than 2, s(0.01, 0) is fluctuated. However, as N approaches 5, the fluctuation is attenuatedand it converges to s ≈ 0.1. A relative error tolerance of 10−10 is used for the simulations where the relativeerror represents the change of solutions divided by a maximum solution at each iteration. This exemplifiesthe accuracy of the present analytic solution.

Another validation of the present analytical solution is made by comparing it with the earlier work ofnumerical solution (Bae et al., 2000), where the Mach number is 0.4 and Tw/T0 = 0.8. The contours of uvelocity are displayed in Figure 3 for λ/δ = 5.4, A = −0.0138 and Re = 10 000. Both the analytical andnumerical contours are seen to be similar in the vicinity of the wall. However, a small discrepancy is ob-served in the region y ≥ 0.05 and z ≈ 0.1 apart from the wall. This may be attributed to the coupling effectbetween the momentum and energy equations in the numerical simulation. The contours of temperature arealso compared in Figure 3. The contours of the analytical solution are in excellent agreement with those ofthe numerical simulation. This suggests that the effect of compressibility by the nonunity isothermal tem-perature ratio is negligible (Huang et al., 1995). The comparison is extended in the contours of pressure. Theanalytical and numerical solutions for pressure show different fashions. In particular, the discrepancy in the

Figure 2. Convergence check.


Figure 3. Contours of u velocity, temperature and pressure at the y–z plane from (a) an analytical solution and (b) a numericalsolution. The tic mark spacing is 0.01. (a) The maximums of u velocity, temperature and pressure are 0.02, 0.718 and 0.7894, re-spectively, and the minimums are 0.0, 0.571 and 0.7940, respectively. (b) The maximums are 0.02, 0.731 and 0.7134, respectively,and the minimums are 0.0, 0.589 and 0.7153, respectively.

region from z = 0 to z = 0.06 is significant. This may be caused by the fact that the analytical pressure fieldis directly influenced by the momentum balance, while the numerical pressure field is governed by the idealgas assumption and the energy balance as well as the momentum balance.

Variations of the skin friction coefficient Cf and the Stanton number St in the z direction are examined,where Cf and St are defined as

Cf = µ(∂u/∂y) |w12ρ∞u∞uedge

, (9)

St = h

ρ∞Cpu∞= −k(∂T/∂y) |w

(Tw − T∞)ρ∞Cpu∞. (10)

Here, ∞ represents the free-stream value, uedge is the value at the edge of boundary layer, h is the heattransfer coefficient, Cp is the specific heat at constant pressure and k is the thermal conductivity, respec-


Figure 4. Variations of skin friction coefficient (Cf) and Stanton number (St) in the normalized spanwise direction (z∗) for λ/δ = 7.4and Re = 5000.

tively. In the presence of inflow disturbances, the Reynolds analogy can be checked by examining thechange of Cf and St, i.e., ∆Cf/Cf0 = ∆St/St0. The notations ∆Cf and ∆St represent ∆Cf = Cf −Cf0 and∆St = St − St0, respectively. The subscript ‘0’ denotes the value without inflow disturbances. The rela-tion ∆Cf/Cf0 = ∆St/St0 is based on the Reynolds analogy in a flat plate boundary layer (dp/dx = 0).However, it is reported (So, 1994) that this relation holds, even though the pressure gradient is not zeroin the stagnation-region flow (dp/dx = 0). To verify whether the relation is satisfied in the stagnation-region flow under small inflow disturbances, the plot of ∆Cf/Cf0 and ∆St/St0 is illustrated againstz∗ = 2π/λ in Figure 4 at A = 0.0002 and 0.0005 with λ/δ = 7.4 and Re = 5000. As shown in Fig-ure 4, the profiles of ∆Cf are sinusoidal for all cases. As the magnitude increases from A = 0.0002 toA = 0.0005, the profiles are more deviated from a sinusoid. However, more deviations from a sinusoidare observed for all cases of ∆St, compared with the profiles of ∆Cf. In particular, the average, max-imum and minimum values of ∆St are larger than those of ∆Cf. The following relation is observed:∆St/St0 ≈ 9.7∆Cf/Cf0 in an average sense, which is against the expectation of ∆Cf/Cf0 ≈ ∆St/St0. It isrecalled in the work of Bae et al. (2000) that the length scale in the present inflow disturbances belongs tothe “detached amplifying” regime. In the regime the streamwise vorticity is significantly amplified by thesmall inflow disturbances.

3.1. Reynolds Analogy Criterion

The Reynolds analogy criterion is made by examining the mean momentum and energy equations. The meanmomentum equation is expressed as

∂u u

∂x+ ∂u v

∂y= −∂ p

∂x+ ∂

∂x

(τ11 −u′u′

)+ ∂

∂y

(τ12 −u′v′

), (11)

where τ11 = 2/Re(∂u/∂x) and τ12 = 1/Re(∂u/∂y +∂v/∂x), respectively. Note that the overbar denotes anaverage over the spanwise direction (z). The variance notation ( )′ represents the velocity fluctuations inresponse to the inflow disturbances. The mean energy equation is

∂u T

∂x+ ∂v T

∂y= ∂

∂x

(1

RePr

∂T

∂x−u′T ′

)+ ∂

∂y

(1

RePr

∂T

∂y−v′T ′

). (12)


By substituting the trigonometric expansions in (5) into (11) and (12), and then integrating from theinflow boundary, the following are obtained:

y∫y=1

2sm(y)2 dy + (sm(y)vm(y)− sm(1)vm(1)) =y∫

y=1

dy −y∫

y=1

N∑n=1

s2an dy − 1

2

N∑n=1

sanvan + 1

Re

∂u

∂y, (13)

y∫y=1

sm(y)Tm(y) dy + (sm(y)Tm(y)− sm(1)Tm(1)) = −1

2

y∫y=1

N∑n=1

sanTan dy − 1

2

N∑n=1

van Tan + 1

RePr

∂T

∂y.

(14)

Here, the integration domain ‘y = 1’ represents the inflow boundary. The integration is taken from y = 1to a certain location y close to the wall, where the direction of integration is opposite to the y axis. The limitof the sum is N = 5.

In the present study the changes (∂u/∂y −∂u0/∂y) and(∂T/∂y −∂T0/∂y

)are taken into consideration as

an indicator of the validity of the Reynolds analogy. These are normalized by the “wall” values (∂u0/∂y)|wand (∂T0/∂y)|w, respectively. The subscript w denotes the wall. The normalized changes for ∂u/∂y and∂T/∂y are represented as(

∂u

∂y− ∂u0

∂y

)/∂u0

∂y

∣∣∣∣w

=y∫

y=1

5∑n=1

s2an dy

/∂u0

∂y

∣∣∣∣w︸︷︷︸

M1

+ 1

2

5∑n=1

sanvan

/∂u0

∂y

∣∣∣∣w︸︷︷︸

M2

+y∫

y=1

2(

s2m(y)− s2

0(y))

dy

/∂u0

∂y

∣∣∣∣w︸︷︷︸

M3

+(sm(y)vm(y)− s0(y)v0(y))− (sm(1)vm(1)− s0(1)v0(1))/

∂u0

∂y

∣∣∣∣w︸︷︷︸

M4

,

(15)

(∂T

∂y− ∂T0

∂y

)/∂T0

∂y

∣∣∣∣w

= 1

2

y∫y=1

5∑n=1

san Tan dy

/∂T0

∂y

∣∣∣∣w︸︷︷︸

E1

+ 1

2

5∑n=1

van Tan

/∂T0

∂y

∣∣∣∣w︸︷︷︸

E2

+y∫

y=1

(sm(y)Tm(y)− s0(y)T0(y)) dy

/∂T0

∂y

∣∣∣∣w︸︷︷︸

E3

+(vm(y)Tm(y)−v0(y)T0(y))− (vm(1)Tm(1)−v0(1)T0(1))/

∂T0

∂y

∣∣∣∣w︸︷︷︸

E4

.

(16)

The marked terms on the right-hand side of (15) represent, respectively, the stress by the fluctuations as-sociated with u′2 (M1), the stress by the fluctuations u and v (M2), the change of mean momentum by a mass


flowing in the x direction from its value without disturbances (M3) and the deviation of mean momentum byu and v (M4). In a manner similar to (15), E1, E2, E3 and E4 in (16) represent the fluctuations of heat flux inthe x direction, the fluctuations in the y direction, the mean deviation of heat flux in the x direction and themean deviation in the y direction, respectively.

A parametric study is performed by varying the magnitude (A) and the Reynolds number (Re) atλ/δ = 10.6. Table 1 shows a set of the parameters conducted in the present study. The relative magnitudesof the terms in (15) and (16) are compared to gauge the contribution of each term to (∂u/∂y −∂u0/∂y) and(∂T/∂y −∂T0/∂y

). Figure 5 shows their profiles for two cases, case B (Re = 10 000 and A = 0.00075) and

case H (Re = 10 000 and A = 0.003). Here, case B represents a weak disturbance, while case H representsa strong disturbance. The horizontal axis is represented by a self-similar coordinate η = y

√Re. The vertical

axis has two scales: ∆Cf/Cf0 and ∆St/(St0 ·a), where a is a limiting factor as obtained earlier, a = 9.7. Here,(∂u/∂y −∂u0/∂y) /∂u0/∂y|w corresponds to ∆Cf/Cf0 and

(∂T/∂y −∂T0/∂y

)/ (∂T0/∂y|w ·a) to ∆St/(St0 ·

a), respectively. Note that the momentum balance and energy balance are similar for a small disturbance(case B), while they are significantly different for a large disturbance (case H). This demonstrates that themomentum and energy balances are significantly influenced by the strength of inflow disturbances.

A global inspection of the budgets in Figure 5 reveals that M3+M4 almost balances with M2. How-ever, it gradually deviates close to the wall, i.e., M3+M4 = M2 from η ≈ 2 to the wall. The momentumis transferred from M3+M4 to M2 in the region η ≥ 2. However, to satisfy the no-slip condition at the

Table 1. Cases conducted in the present study.

Cases λ/δ Re A×10−3

A 10.6 5000 0.75B 10.6 10 000 0.75C 10.6 20 000 0.75D 10.6 5000 1.5E 10.6 10 000 1.5F 10.6 20 000 1.5G 10.6 5000 3.0H 10.6 10 000 3.0I 10.6 20 000 3.0

Figure 5. Momentum and energy budgets for (a) case B and (b) case H.


wall, the momentum recovers from M2 to M3+M4 in the region η ≤ 2. The unrecovered momentum yieldsthe increase of (∂u/∂y −∂u0/∂y) /(∂u0/∂y)|w near the wall. This brings forth the positive wall value of(∂u/∂y −∂u0/∂y) /(∂u0/∂y)|w. A similar trend is seen in the energy budget. E3+E4 balances with E2 out-side the wall region (0 ≤ η ≤ 2). Similarly to the momentum,

(∂T/∂y −∂T0/∂y

)/(∂T0/∂y|w ·a) is positive

at the wall due to the wall constraint. Figure 5 illustrates how the breakdown evolves as the strength of theinflow disturbances increases. Note that for case H, E2 and E3+E4 develop much stronger than M2 andM3+M4 for η ≥ 2.

The correlations of M2 and E2 with the changes of ∂u/∂y and ∂T/∂y are illustrated in Figure 6 for threecases E, F and I. As seen in Figure 6, the peak values of M2 and E2 are closely correlated with the wallvalues of (∂u/∂y −∂u0/∂y) /(∂u0/∂y)|w and

(∂T/∂y −∂T0/∂y

)/((∂T0/∂y)|w ·a). As M2 and E2 increase,

the unrecovered momentum and heat flux in the vicinity of the wall increase. Furthermore, the wall value of(∂T/∂y −∂T0/∂y

)/((∂T0/∂y)|w ·a) is slightly larger than that of (∂u/∂y −∂u0/∂y) /(∂u0/∂y)|w. Based on

the findings from Figures 5 and 6, the breakdown of Reynolds analogy is directly related to M2 and E2 thatshould balance with the mean distortions M3+M4 and E3+E4, respectively. This suggests that the flow andtemperature fields are disturbed differently by the inflow disturbances, i.e., the different evolutions of M2 andE2 cause the different values of ∆Cf/Cf0 and ∆St/(St0 ·a).

The present analysis is extended to the behaviors of M3 and M4 for momentum, and E3 and E4 for energyin Figure 7. M3 is similar to E3, i.e., the roles of M3 and E3 on the behaviors of M2 and E2 are insignificant.However, M4 is different from E4. The behaviors of E4 are broader than those of M4. This means that the in-fluences of M4 and E4 on M2 and E2 are substantial. As the wall values of

(∂T/∂y −∂T0/∂y

)/((∂T0/∂y)|w ·

Figure 6. (a) M2 and (∂u/∂y −∂u0/∂y) /(∂u0/∂y)|w and (b) E2 and(∂T/∂y −∂T0/∂y

)/((∂T0∂y)|w ·a) for cases E, F and I.


0 2 4 6 8 10-0.02

0

0.02

0.04

0.06M3M4

a)

b)

case E

case E

case F

case F

case I

case I

case I

case F

case E

case E

case I

case F

Figure 7. (a) M3 and M4 and (b) E3 and E4 for cases E, F and I.

a) in Figure 6(b) increase, the absolute peak values of E4 also increase for all cases. However, the responsesof M4 to the increase of (∂u/∂y −∂u0/∂y) /(∂u0/∂y)|w are seen to be less sensitive compared with those ofE4. It is interesting to note that M4 and E4 contain vm in (15) and (16) while M3 and E3 include sm . Asa result, vm is related to the breakdown of similarity between M4 and E4 while sm is associated with thesimilarity between M3 and E3.

The foregoing analysis indicates that the breakdown is linked to the different evolutions of M2 and E2.Furthermore, the mean deviations of the momentum and energy transfers are more influenced by vm , i.e., M4and E4 are responsible for the different evolutions of M2 and E2. To see the influence of vm , the contourplots of u and T are illustrated in Figure 8. The profile of ∆vm = vm −v0 is also included. The + and −signs represent the cores of counterclockwise and clockwise rotating streamwise vortices (ωx), respectively.It is seen that the temperature contours in the range of the pulling motion are closely correlated with ∆vm :a mushroom shape is captured in the temperature contours. However, the dependence of u on ∆vm is lesssignificant in the pulling motion. Recall that E2 and E4 in Figures 6 and 7 are broader than M2 and M4.

3.2. Analysis Using Disturbance Correlations

As exhibited in Figure 1, streamwise vorticity is amplified by the mean strain rate as the flow approachesthe wall. The amplified vorticity resides near the edge of the boundary layer and disturbs the flow and tem-perature fields. These reveal that streamwise vorticity plays a key role in the structures. A fluctuating velocitytransport equation is employed for the detailed investigation of Reynolds analogy breakdown in the presence


Figure 8. Schematic diagram for u velocity and temperature with the plot of ∆vm against y.

of inflow disturbances. The fluctuating velocity transport equation in vorticity form is expressed as (Lele,1992)

∂u′i

∂t+ ∂

∂xi

(p′

ρ+uku′

k + 1

2u′

ku′k − 1

2〈u′

ku′k〉)

= (u ×ω)i + (u′ ×Ω)i + (u′ ×ω)i −〈u′ ×ω〉i − 1

Re(×ω)i, (17)

where 〈·〉 represents an ensemble average over all the disturbance modes. Since the flow is steady, the firstterm is neglected. The fluctuating velocity equation in the y direction is selected and integrated from theinflow boundary:

(p′

ρ+uku′

k + 1

2u′

ku′k − 1

2

⟨u′

ku′k

⟩)−(

p′

ρ+uku′

k + 1

2u′

ku′k − 1

2

⟨u′

ku′k

⟩)y=1

=y∫

y=1

(u ×ω)2 + (u′ ×Ω)2 + (u′ ×ω)2 −〈u′ ×ω〉2 − 1

Re(×ω)2 dy . (18)

By multiplying both sides of (18) by u′ and T ′ and averaging over the spanwise direction with the self-similarity assumption, the following equations are obtained:

v⟨s′v′⟩+ ⟨

p′s′⟩︸︷︷︸uv1

+1

2

⟨

s′v′v′⟩︸︷︷︸uv2

+ ⟨s′w′w′⟩︸︷︷︸

uv3

−

⟨s′(

p′(1)+v(1)v′(1)+ 1

2v′(1)v′(1)+ 1

2w′(1)w′(1)

)⟩

(19)

=⟨

s′y∫

y=1

w′ω1, dy

⟩︸︷︷︸

uv4

−⟨

s′y∫

y=1

1

Re

(∂ω1

∂z− ∂ω3

∂x

)dy

⟩︸︷︷︸

uv5

,


v⟨v′T ′⟩+ ⟨

p′T ′⟩︸︷︷︸vt1

+1

2

⟨

v′v′T ′⟩︸︷︷︸vt2

+ ⟨w′w′T ′⟩︸︷︷︸

vt3

−

⟨T ′

(p′(1)+v(1)v′(1)+ 1

2v′(1)v′(1)+ 1

2w′(1)w′(1)

)⟩

(20)

=⟨

T ′y∫

y=1

w′ω1 dy

⟩︸︷︷︸

vt4

−⟨

T ′y∫

y=1

1

Re

(∂ω1

∂z− ∂ω3

∂x

)dy

⟩︸︷︷︸

vt5

.

In the above, s′ denotes u′/x and ω is the vorticity fluctuation, respectively. Equations (19) and (20) de-scribe the relation of 〈s′v′〉 and 〈v′T ′〉 with other terms. Each term in (19) has its corresponding term in(20). Assuming v(1) ≈ 1.0 and p′(1),w′(1) v′(1), the terms related to the inflow boundary values areapproximated as ⟨

s′(

p′(1)+v(1)v′(1)+ 1

2v′(1)v′(1)+ 1

2w′(1)w′(1)

)⟩≈ ⟨

s′v′(1)⟩,⟨

T ′(

p′(1)+v(1)v′(1)+ 1

2v′(1)v′(1)+ 1

2w′(1)w′(1)

)⟩≈ ⟨

T ′v′(1)⟩. (21)

This shows that the inflow boundary term represents a reference value based on the magnitude of inflowdisturbances.

The breakdown mechanism can be analyzed by considering the physical meanings of the five terms in(19) and (20). The uv1 and vt1 terms are related to the pressure fluctuations (p′); the uv2 and vt2 terms areselated to the wall-normal velocity fluctuation (v′2); the uv3 and vt3 terms are selated to the spanwise vel-ocity fluctuations (w′2); the uv4 and vt4 terms are selated to the wall-normal component of the Lamb vector(Wu et al., 1999), i.e., I = u×ω; and the uv5 and vt5 terms are selated to the diffusion of wall-normal vel-ocity fluctuations, i.e., (1/Re)(∂ω1/∂z −∂ω3/∂x) = −(1/Re)∇2v′. The pressure fluctuations are genericallyassociated with the momentum balances by the so-called “redistribution.” The wall normal velocity fluctu-ations (v′2) are closely connected with the “wall blocking” effect (Hunt and Graham, 1978; Hancock andBradshaw, 1983). In the present stagnation-region flow, this blocking effect by the solid wall is expected tobe significant. The normal component of the Lamb vector is denoted as I2 = wω1 −uω3. However, the sec-ond term −uω3 in I2 is related by

∫ −uω3 dy = 12 u′u′ in (17). Accordingly, only the first term wω1 is used

to represent the uv4 and vt4 terms in (19) and (20). When the streamwise vortex is symmetric in the y di-rection and w is asymmetric,

∫wω1dy is zero. In this study the streamwise vortex is not symmetric in the y

direction and w is not uniform due to the no-slip wall. As the streamwise vortex tube becomes strong by thevortex stretching near the wall, the

∫wω1dy term increases.

To look into the responses of the above-stated five terms to the inflow disturbances, profiles of the fiveterms close to the wall are displayed in Figure 9 for two disturbances (cases A and I). Again, case A repre-sents a weak disturbance at Re = 5000 and A = 0.00075 while case I is a strong disturbance at Re = 20 000and A = 0.003. The uvi terms (i = 1, 2, 3, 4, 5) are normalized by v′

max and s′max, where the subscript max

denotes the maximum value. With the same analogy, the vti terms are normalized by v′max and T ′

max. Fora weak disturbance (case A), the pressure fluctuations (uv1, vt1) and the diffusion (uv5, vt5) respond toinflow disturbances significantly compared with the other three terms, as shown in Figure 9(a). A close-up view for the three terms in Figure 9(a) discloses that the responses of the Lamb vector (uv4, vt4) arevery small in magnitude. Moreover, their shapes are qualitatively different from the other two terms, i.e., theprofile of uv4 is different from that of vt4. As the inflow disturbances are stronger (case I), the pressure fluc-tuations (uv1, vt1) and the diffusion (uv5, vt5) are still dominant as shown in Figure 9(b). It is remarkableto find that the wall blocking terms (uv2, vt2) are significantly sensitive to the strong inflow disturbances ina negative direction.

As the strength of inflow disturbances increases from case A to case I, some terms show a signifi-cant change in the profiles of momentum and energy balances, while other terms remain unchanged.A minor change is observed in the diffusion term (uv5, vt5). A closer inspection of the changes of each


0 2 4 6 8 100

0.01

0.02

0.03

0.04

0.05

momentumenergy

a) b)

vt1

vt1

vt5vt5

vt3

vt2

vt4

vt3

vt2

vt4

uv1

uv5uv5

uv1

uv3

uv4

uv2

uv2

uv3

uv4

η η

case A

case A

case I

case I

Figure 9. Comparison of the terms in (19) and (20) for (a) case A and (b) case I.

term indicates that the peak values of the pressure fluctuations (uv1, vt1) decrease with increasing thestrength of inflow disturbances. On the other hand, vt1 has a second peak near η ≈ 6 while uv1 de-creases monotonically. It is interesting to detect that the peak position of vt2 moves from η ≈ 2 to η ≈ 6as the strength increases. This suggests that the pressure fluctuations behave similarly to the wall block-ing with the evolution of a streamwise vortex. When we look at the change of the Lamb vectors (uv4,vt4), they are completely dissimilar and their absolute values are the smallest among the five terms. Al-though their magnitudes are negligible, the Lamb vectors are closely connected with the evolution ofa streamwise vortex (Wu et al., 1999). Note that the Lamb vectors change significantly near η ≈ 6. Onthe contrary, as mentioned earlier, the changes of the diffusion (uv5, vt5) and the spanwise fluctuations(uv3, vt3) are insignificant with increasing the strength of inflow disturbances. A global examination ofFigure 9 indicates that the three terms, i.e., the pressure fluctuations, the wall blocking and the Lamb vec-tors respond significantly to the change of inflow disturbances. In addition, these terms show differentpatterns between the momentum and energy transfers, especially near η ≈ 6. This demonstrates that thepressure fluctuations, the wall blocking and the Lamb vectors may be responsible for the breakdown of theReynolds analogy.

As listed in Table 1, nine cases of inflow disturbances are chosen by varying Re and A at λ/δ = 10.6.Based on these selected cases, the influences of Re and A on the aforementioned terms are evaluated. Fourcases are selected in Table 2, which belong to the regime of weak inflow disturbances. Cases in the regimeof strong inflow disturbances are listed in Table 3. The values in Tables 2 and 3 are obtained by calculatingratios of the peak values. A general trend in Table 2 is that as Re and A increase, the absolute peak valueof each term increases. For the pressure fluctuations and the diffusion, the peak values are proportional toA2 and Re. On the other hand, the wall blocking and the spanwise fluctuations are augmented by A4 andRe3. However, the absolute values of the pressure fluctuations and the diffusion are substantially larger than


Table 2. Peak value ratios for weak inflow disturbances.

Pressure Wall SpanwiseRatios fluctuations blocking fluctuations Diffusion

Case B / case A 1.9 7.7 7.1 2.1Case C / case B 1.8 8.0 5.9 2.1Case D / case A 3.9 16.4 15.0 4.0Case G / case D 3.7 17.0 12.4 3.9

Table 3. Peak value ratios for strong inflow disturbances.

Pressure Wall SpanwiseRatios fluctuations blocking fluctuations Diffusion

Case H / case G 1.5 7.0 3.1 1.9Case I / case H 0.92 1.8 2.3 1.3Case F / case C 3.1 14.9 6.5 3.7Case I / case F 1.9 3.8 4.8 2.5

those of the wall blocking and the spanwise fluctuations. Accordingly, 〈s′v′〉 and 〈v′T ′〉 are proportional to(A

√Re)2.

The peak-value ratios for strong inflow disturbances are listed in Table 3. Contrary to the prior weak in-flow disturbances, the ratios decrease significantly as Re and A increase. Furthermore, the peak values ofthe pressure fluctuations and the wall blocking are saturated rapidly compared with those of the spanwisefluctuations and the diffusion. Note that the peak value of the pressure fluctuations in case H is larger thanthat in case I. This is attributed to the fact that the pressure fluctuations in “case I” have a second peak nearη = 6 (see Figure 9) while those in case H do not. As the second peak develops, the profile of the pressurefluctuations is widened so that the relative importance of the first peak decreases. In general, it is shown thatthe dependency of A

√Re is also maintained in Table 3.

A detailed response of the Lamb vectors to the different cases of inflow disturbances is exhibited in Fig-ure 10. This is because the Lamb vectors show a dissimilar behavior eminently, compared with other terms.As remarked earlier, cases A, B and C in Figure 10 belong to the regime of weak inflow disturbances. InFigure 10(a) the peak values increase with increasing Re near η = 1. The second peaks are weak, i.e., thefluctuations decay as η → ∞. It is estimated that the peak ratios are proportional to A4 and Re3, which issimilar to the trend of the wall blocking and the spanwise fluctuations. However, the second peaks in Fig-ure 10(b) become substantial as Re increases. Since the Lamb vectors contain both vorticity and velocityvectors, many studies have been made to identify the nonlinearity effect in fluid dynamics (Tsinober, 1990;Shtilman, 1992; Wu et al., 1999). As mentioned earlier, the Lamb vectors play a key contribution to the

Figure 10. Comparison of the Lamb vectors for (a) momentum and (b) energy with the change of the Reynolds number atA = 0.00075.


n=1

a)

n=2

b)

n=3

c)η η η

pressurefluctuations

wallblocking

Lamb

Figure 11. (a) The pressure fluctuations (uv1 and vt1), (b) the wall blocking (uv2 and vt2) and (c) the Lamb vectors (uv4 and vt4)for case I (λ/δ = 10.6, Re = 20 000, A = 0.003) with the change of the mode number n.

breakdown of the Reynolds analogy, although its magnitude is small compared with those of the pressurefluctuations and the wall blocking.

Efforts are extended to see the detailed structures close to the wall. Toward this end, flow modes of thethree dominant terms up to n = 3 are displayed in Figure 11, where n denotes the normalized wave number.For the pressure fluctuations (uv1, vt1), the first mode (n = 1) is dominant among the three modes. As n in-creases, a ‘redistribution’ is shown by the pressure fluctuations. At n = 1, vt1 is larger than uv1. However,it is reversed at n = 2 and n = 3. Contrary to the pressure fluctuations, the wall blocking (uv2, vt2) showsa different behavior, i.e., the signs of uv2 and vt2 at n = 2 and n = 3 are the same as those at n = 1. Further-more, the absolute value of vt2 is still larger than that of uv2. This means that the contributions from smallerscales are supportive to larger scales in the wall blocking. For the Lamb vectors (uv4, vt4), the shapes of uv4are qualitatively different from those of vt4. Compared with the above two terms, the absolute peak valuesof the Lamb vectors are gradually attenuated as n increases. A closer inspection of the profiles in Figure 11discloses that significant changes are shown near η = 6 for all vti (i = 1, 2, 4) terms. This may be linked withthe aforementioned evolution of a streamwise vortex at the location.

The correlations of s′ and T ′ with the five terms (p′, v′2,∫

w′ω1dy, w′2,∫(1/Re)(∂ω1/∂z −∂ω3/∂x) dy)

are examined by plotting the y–z contours in Figure 12 at n = 1. Case I is selected, i.e., λ/δ = 10.6,A = 0.003 and Re = 20 000. Solid lines represent positive values and dotted lines denote negative ones. They and z coordinates are multiplied by

√Re so that their scales are the same as the self-similar coordinate. As

seen in Figure 12(b), T ′ extends its contour-structure up to η = 8, while s′ ends near η = 4. It is interestingto see that the contours of w′2 and

∫(1/Re)(∂ω1/∂z −∂ω3/∂x) dy have peaks near η = 2, similar to those

of s′ . The second peak of w′2 is located further away from the peak locations of s′ and T ′ and its absolutevalue is negligible compared with that of the first peak. On the other hand, the contours related to the threedominant terms (p′, v′2,

∫w′ω1dy) have peaks near η = 6. It is evident that the contours of T ′ have a neck-

type structure near η = 6, while the contours of s′ do not. This suggests that T ′ correlates more closely withthe three terms near η = 6 than s′. The contour values of w′2 and

∫(1/Re)(∂ω1/∂z −∂ω3/∂x) dy are small

near η = 6. Recall that the pressure fluctuations and the wall blocking are located in the left-hand side in (19)


Figure 12. Contours of (a) s′, (b) T ′, (c) p′, (d) v′2, (e)∫

w′ω1dy, (f) w′2, (g)∫(1/Re) (∂ω1/∂z −∂ω3/∂x) dy for case I (λ/δ = 10.6,

Re = 20 000, A = 0.003) and n = 1. Both y and z coordinates are multiplied by√

Re. Extreme values are (a) 0.0839, (b) 0.000193,(c) 0.00195, (d) 0.00195, (e) 0.000813, (f) 0.000827 and (g) 0.000649. Solid lines represent positive values and dotted lines denotenegative values. Twenty contour levels are used.

and (20) while the Lamb vectors are in the right-hand side. Considering the sign notations, the wall block-ing and the Lamb vectors contribute positively to 〈s′v′〉 and 〈v′T ′〉, while the pressure fluctuations contributenegatively for 0 ≤ η ≤ 10.0. Moreover, p′ has a strong structure near the wall, which is also observed in thecontours of w′2 and

∫(1/Re)(∂ω1/∂z −∂ω3/∂x) dy. Consequently, the pressure fluctuations balance with the

spanwise fluctuations and the diffusion in the region. The contours for n = 2 are shown in Figure 13. Thelegend is the same as that of Figure 12. Contrary to Figure 12, the contours of s′ extend up to η = 6. A sec-ondary structure of T ′ exists near η = 8. The peak locations of the three dominant terms (p′, v′2,

∫w′ω1dy)

are located at η ≈ 8. A detailed examination of the profiles in Figure 13 discloses that contributions of thepressure fluctuations and the Lamb vectors to 〈s′v′〉 and 〈v′T ′〉 are exchanged. The pressure fluctuations andthe wall blocking contribute positively to 〈s′v′〉 and 〈v′T ′〉, while the Lamb vectors contribute negativelyfor n = 2.

Finally, the y–z contours of (a) s′v′, (b) s′, (c) v′T ′ and (d) T ′ for case I are displayed in Figure 14. Boththe y and z coordinates are multiplied by

√Re. The contours of v′T ′ have a similar shape to those of s′v′

excluding the region 5 ≤ y ≤ 10, where v′T ′ shows a stronger contour-structure than s′v′. This is attributed


Figure 13. Contours of (a) s′, (b) T ′, (c) p′, (d) v′2, (e)∫

w′ω1 dy, (f) w′2, (g)∫(1/Re) (∂ω1/∂z −∂ω3/∂x) dy for case I (λ/δ = 10.6,

Re = 20 000, A = 0.003) and n = 2. Both y and z coordinates are multiplied by√

Re. Extreme values are (a) 0.0347, (b) 0.000979,(c) 0.000440, (d) 0.00245, (e) 0.000442, (f) 0.00129 and (g) 0.000296. Solid lines represent positive values and dotted lines denotenegative values. Twenty contour levels are used.

to the fact that T ′ has a stronger contour-structure for 5 ≤ y ≤ 10 than s′, as shown in Figures 14(b),(d). Thisdemonstrates that v′T ′ has a larger spanwise-averaged value than s′v′ in the region, which may be related tothe breakdown of the Reynolds analogy.

4. Conclusions

An analytical method with the assumption of self-similarity is applied to the steady incompressiblestagnation-region flow and heat transfer in the presence of inflow disturbances. The magnitude (A) andthe Reynolds number (Re) at a fixed length scale λ/δ = 10.6 are varied to investigate the mechanism re-sponsible for the Reynolds analogy breakdown. Two approaches are employed for the detailed mechanismanalysis. It is found by the Reynolds analogy criterion that (∂u/∂y − ∂u0/∂y)/(∂u0/∂y)|w and (∂T/∂y −∂T0/∂y)/((∂T0/∂y)|w ·a) are dissimilar significantly for the stronger inflow disturbances, which confirmsthe Reynolds analogy breakdown. The different evolutions of (∂u/∂y −∂u0/∂y)/(∂u0/∂y)|w and (∂T/∂y −


Figure 14. Contours of (a) s′v′, (b) s′, (c) v′T ′ and (d) T ′ for case I (λ/δ=10.6, Re=20000, A=0.003). Both y and z coordinatesare multiplied by

√Re. (a) maximum value: 0.000232, minimum value: -0.00320 and contour gap: 0.000320; (b) maximum value:

0.0695, minimum value: -0.129 and contour gap: 0.0129; (c) maximum value: 0.00000135, minimum value: -0.0000157 and con-tour gap: 0.00000157; (d) maximum value: 0.000172, minimum value: -0.000340 and contour gap: 0.0000340. Solid lines representpositive values and dotted lines denote negative values.

∂T0/∂y)/((∂T0/∂y)|w ·a) are attributed to the difference between M2 and E2. Furthermore, the behaviors ofM4 and E4 are significantly different. From the analysis using disturbance correlations, it is demonstratedthat the pressure fluctuations, the wall blocking and the Lamb vectors are responsible for the breakdown ofthe Reynolds analogy. The pressure fluctuations display (A

√Re)2 dependency for the weaker inflow dis-

turbances. For the stronger inflow disturbances, they are saturated rapidly. The “redistribution” of pressurefluctuations is shown with the change of the mode number. The Lamb vectors show a significant dissimi-larity between the momentum and energy balances even for the weaker inflow disturbances. However, theirabsolute values of the peaks are smaller than those of the pressure fluctuations and the wall blocking. A com-mon finding of the three terms is that their profiles are changed significantly near η = 6, which is linked withthe evolution of a streamwise vortex at the location. The y–z contour plots of s′, T ′, p′, v′2,

∫w′ω1dy, w′2

and∫(1/Re)(∂ω1/∂z −∂ω3/∂x)dy reveal that the peaks of p′, v′2,

∫w′ω1dy are located where the T ′ peak

is located. Likewise, the peaks of w′2 and∫(1/Re)(∂ω1/∂z −∂ω3/∂x) dy are located where the s′ peak is

located.

Appendix

The trigonometrically expanded equations are listed as follows.

Continuity equation:

sm +v′m +

∞∑n=1

(san +v′an + knwbn) cos(kn z)+ (sbn +v′

bn −knwan) sin(kn z) = 0 . (22)


X-momentum equation:

s2m +2

∞∑n=1

smsan cos(kn z)+ smsbn sin(kn z)

+vms′m +

∞∑n=1

(vms′an + s′

mvan) cos(kn z)+ (vms′bn + s′

mvbn) sin(kn z)

+∞∑

l=1

∞∑m=1

1

2(salsam +vals

′am +walsbmkm) cos((kl −km)z)+cos((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(salsbm +vals

′bm −walsamkm) sin((km −kl)z)+ sin((kl +km )z)

+∞∑

l=1

∞∑m=1

1

2(sblsam +vbls

′am +wblsbmkm) sin((kl −km)z)+ sin((kl +km )z)

+∞∑

l=1

∞∑m=1

1

2(sblsbm +vbls

′bm −wblsamkm) cos((kl −km)z)−cos((kl +km)z)

−1− 1

Re

s′′

m +∞∑

n=1

(s′′

an −k2nsan

)cos(kn z)+

(s′′

bn −k2nsbn

)sin(kn z)

= 0 . (23)

Y-momentum equation:

vmv′m +

∞∑n=1

(vmv′an +v′

mvan) cos(kn z)+ (vmv′bn +v′

mvbn) sin(kn z)

+∞∑

l=1

∞∑m=1

1

2(valv

′am +walvbmkm) cos((kl −km)z)+cos((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(valv

′bm −walvamkm) sin((km −kl)z)+ sin((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(vblv

′am +wblvbmkm) sin((kl −km )z)+ sin((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(vblv

′bm −wblvamkm) cos((kl −km)z)−cos((kl +km)z)

+ p′m +

∞∑n=1

p′an cos(kn z)+ p′

bn sin(kn z)

− 1

Re

v′′

m +∞∑

n=1

(v′′

an −k2nvan

)cos(kn z)+

(v′′

bn −k2nvbn

)sin(kn z)

= 0 . (24)


Z-momentum equation:

∞∑n=1

vmw′an cos(kn z)+vmw′

bn sin(kn z)

+∞∑

l=1

∞∑m=1

1

2(valw

′am +walwbmkm) cos((kl −km)z)+cos((kl +km )z)

+∞∑

l=1

∞∑m=1

1

2(valw

′bm −walwamkm) sin((km −kl)z)+ sin((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(vblw

′am +wblwbmkm) sin((kl −km)z)+ sin((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(vblw

′bm −wblwamkm) cos((kl −km)z)−cos((kl +km )z)

+∞∑

n=1

kn pbn cos(kn z)−kn pan sin(kn z)

− 1

Re

∞∑n=1

(w′′

an − k2nwan

)cos(kn z)+

(w′′

bn −k2nwbn

)sin(kn z)

= 0 . (25)

Energy equation:

vm T ′m +

∞∑n=1

(vmT ′an + T ′

mvan) cos(kn z)+ (vm T ′bn + T ′

mvbn) sin(kn z)

+∞∑

l=1

∞∑m=1

1

2(valT

′am +walTbmkm) cos((kl −km)z)+cos((kl +km)z)

+∞∑

l=1

∞∑m=1

1

2(valT

′bm −walTamkm) sin((km −kl)z)+ sin((kl +km )z)

+∞∑

l=1

∞∑m=1

1

2(vblT

′am +wblTbmkm) sin((kl −km)z)+ sin((kl +km )z)

+∞∑

l=1

∞∑m=1

1

2(vblT

′bm −wblTamkm) cos((kl −km)z)−cos((kl +km)z)

− 1

RePr

T ′′

m +∞∑

n=1

(T ′′

an −k2n Tan

)cos(kn z)+

(T ′′

bn −k2n Tbn

)sin(kn z)

= 0 . (26)

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kaistflow.kaist.ac.kr/upload/paper/2001/2001____tacfd_14...the governing equations and collect terms...

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