9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, June 5-8 2006, San Francisco, California
Turbulent Navier-Stokes Simulations of Heat Transfer
with Complex Wall Temperature Variations
Eric C. Marineau ∗
and
Joseph A. Schetz †
Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061
and
Reece E. Neel ‡
AeroSoft, Inc., Blacksburg, VA, 24061
Numerical simulations focusing on convective heat transfer with complex wall tempera-ture variations, as well as conjugate heat transfer problems involving solids are performed.Simulations are made using the GASP Computational Fluid Dynamics (CFD)code, whichsolves the Reynolds-Averaged Navier-Stokes equations (RANS). GASP has been modifiedby adding a solid heat conduction solver, enabling the coupling of RANS with the heatequation for an isotropic solid. Validation cases involving convective heat transfer are con-sidered using both one- and two-equations turbulence models. Predicted Stanton numbersfor low Mach number turbulent boundary layers closely agree with experimental resultsfrom Reynolds et al. 1,2 for a constant wall temperature, a step and a double pulse inwall temperature. Simulations of supersonic boundary layers with a step in wall temper-ature are also performed. Good agreement is found for velocity and temperature profileswhen compared with measurements from Debieve et al. 3 as well as for the skin frictionand Stanton number. The flow field and wall temperature distribution inside a supersoniccooled nozzle is computed using a new conjugate heat transfer algorithm in GASP. Resultsagree well with measurements from Back et al. 4
Nomenclature
α Cebeci and Smith closure coefficientαt Turbulent thermal diffusivityβ∗ Wilcox k − ω closure coefficientδ Boundary layer thicknessδ∗ Boundary layer displacement thicknessq Heat sourceε Dissipation rate of turbulent kinetic energyη Normalized normal wall distance, η = y/δγ Specific heat ratio, γ = cp/cv
ω Specific dissipation rate of turbulent kinetic energyρ Densityτw Wall shear stressτxy Principal Reynolds shear stress, τxy = µt
∂u∂y
θ Boundary layer momentum thicknessA Face areaa Structural parameterA+ Van Driest damping constantC Law of the wall constantC1 First asymptotic matching constantC2 Second asymptotic matching constantCf Skin friction coefficient, Cf = 2τw/ρeU
2e
cp Specific heat at constant pressurecv Specific heat at constant volumeFkleb Klebanoff intermittency functionk Turbulent kinetic energy or thermal conductivitykt Thermal eddy conductivitylmix Mixing lengthM Mach numberp PressurePr Prandtl number, Pr = cpµ/kf
q Heat fluxr Recovery factorReθ Reynolds’s number based on momentum displacement thicknessSt Stanton number, St = qw/ρeUecp(Tw − Tr)Tr Recovery temperature, Tr = Te + rU2
e /2cp
u, v, w Cartesian velocity componentsu+ Scaled velocity, u+ = u/uτ
uc Van Driest transformed velocityuτ Friction velocity, uτ =
√τw/ρw
V Cell volumey Normal wall distancey+ Turbulent length, y+ = yuτ/νw
Subscripts0 Stagnation or initial valuee Boundary layer edge valuef Fluids Solidw Wall value
I. Introduction
The coupling of heat transfer at a solid/fluid interface is known as conjugate heat transfer (CHT). CHTproblems are commonly found in real-world applications such as turbo-machinery, reentry vehicles, laserirradiation applications, heating ducts and more. In order to properly simulate a conjugate heat transferproblem, a code needs to be able to accurately model the convective heat transfer in the fluid and theconductive heat transfer in the solid.
CHT capabilities have been added to GASP allowing tight coupling of the RANS equations in the fluidto the heat conduction equation in the solid. Since conjugate heat transfer problems involve convective heattransfer at the fluid-solid interface, accurate modeling of convection is critical when solving these problems.Therefore, several cases involving convective heat transfer for low and high Mach number boundary layers
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are presented. These cases serve as validation problems for the GASP fluid flow solver.Three low Mach number cases are considered, namely a heated wall at constant temperature, a step
in wall temperature and a double heat pulse. Numerical results of the Stanton number are compared toexperimental results from Reynolds et al. 1,2
Three supersonic cases are considered which include an adiabatic wall case and a step in the wall tem-perature with ratio of wall to recovery temperature, Tw/Tr, of 1.5 and 2 respectively. Skin friction, Stantonnumber as well as velocity and temperature profiles are compared to experimental results from Debieve et al.3
A single conjugate heat transfer problem is considered, namely an axisymmetric nozzle with an exit machnumber of 2.6 involving heated flow of air and a water cooled wall. The flow field and the wall temperaturedistribution are computed and compared to measurements from Back et al.4
II. GASP Flow Solver
A recent version of GASP, Version 4.3, was used for this study. GASP is a commercial CFD flow solverdeveloped by AeroSoft, Inc.5 It solves the integral form of the time-dependent Reynolds-Averaged Navier-Stokes (RANS) equations in three dimensions.
To model turbulence, GASP has an array of options. These include the Baldwin-Lomax6 algebraic modelin any two logical directions, the one-equation Spalart-Allmaras7 model, two-equation models based uponthe k-ω and k-ε equations, and a second-order Reynolds stress closure model by Wilcox denoted as thestress-ω model.8 The k-ω models include Wilcox’s 19889 and 19988 high Reynolds number models, Wilcox’s1998 low Reynolds number model,8 and Menter’s SST model.10 For the k-ε models, GASP supports a highReynolds number model,11 Chien’s model,12 and the model by Lam and Bremhorst.13 All the above modelssupport user-input, intermittency values for transition modeling. Additional information and validation ofthe turbulence modeling in GASP can be found in Neel at al.14 All single and multi-equation models canbe run uncoupled from the primary flow equations for more efficient CPU times.
GASP uses message passing interface (MPI) in order to run on both shared and distributed memoryplatforms. GASP provides users with a semi-automated domain decomposition in order to take full advan-tage of the parallel capability. For both single or multi-processor jobs, GASP supports full implicit timeintegration.
The turbulent heat fluxes are modeled using the turbulent Prandtl number, Prt, which is assumed to beconstant across the boundary layer. The turbulent heat fluxes are expressed in the following way.
−u′′i T ′′ = αt∂T
∂xi=
νt
Prt
∂T
∂xi(1)
where αt is the turbulent thermal diffusivity, νt is the dynamic eddy viscosity and () denotes Favre averagingand ()′′ the fluctuation of () respectively. In this formulation, a similarity between turbulent heat andmomentum transfer is assumed since the turbulent thermal diffusivity is directly computed from the dynamiceddy viscosity.15 For wall bounded flow, experimental evidence shows that Prt increases close to the wall.However when the Prandtl number is less than one the value of Prt at y+ < 7 or 8 has negligible effecton calculations.16 Therefore for air the constant Prandtl number approximation is reasonable. Kays16
presents a correlation for the variation of Prt across a boundary layer without pressure gradients whichagrees reasonably well with experimental data.
Another approach involves modeling the temperature fluctuations.15,17–19 Complexity and computationalcost are increased as two additional equations for the temperature variance and its dissipation rate need tobe coupled to the turbulence model. This approach is being investigated for future work.
III. The Heat Equation and Conjugate Heat Transfer Modeling
A solver for the three-dimensional heat conduction equation has been added to GASP in order to per-form conjugate heat transfer problems. The time dependent heat transfer equation for an isotropic, solidcontinuum is expressed as,
ρcv∂T
∂t=
∂
∂x(k
∂T
∂x) +
∂
∂y(k
∂T
∂y) +
∂
∂z(k
∂T
∂z) + q (2)
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where ρ is the material density, cv is the specific heat, and k is the thermal conductivity, all of which canbe functions of temperature. The source term q represents an internal energy source. This equation can berecast into integral form as,
ρcv∂
∂t
∫∫∫T dV =
∮A
(k(∇T · n) dA +∫∫∫
q dV. (3)
which lends itself to the finite volume implementation. In the finite volume solver, the unknown (in our casetemperature) is assumed to be volume averaged over each cell in the mesh. This can be expressed as,
T =1V
∫∫∫T dV (4)
where the bar over T indicates a volume average.The diffusive term (heat flux) is computed in the same fashion as the viscous terms used in the Navier-
Stokes solver. This yields a second order accurate central difference formulation for the diffusive flux. Fortime accurate flows, the dual-time stepping algorithm is used. In this situation, the algorithm presented nextis repeated for a set number of cycles in which the heat flux is converged for a given physical time step. Thealgorithm for performing CHT problems is now explained. In the following description, it may be helpful torefer to figure 1 for terminology. In this figure, the fluid dynamics zone (or grid) is on top and the shaded
Figure 1: Schematic of the conjugate heat transfer problem
zone on bottom is the solid material zone. The boundary condition at a zonal boundary face (face commonto both a fluid zone and a solid zone) is:
Tfw = Tsw = Tw (wall temperatures equal)
andqfw = −qsw (heat fluxes equal and opposite)
where
qfw = −kf∂T
∂n
∣∣∣∣fw
qsw = −ks∂T
∂n
∣∣∣∣sw
When a surface heat source is present, the boundary condition becomes
qfw = −qsw − qo
where qo is an additional heat flux specified by the user. The above condition states the conservation ofenergy at the surface.
The algorithm covering one iteration cycle is as follows.
1. The wall temperature along the zonal boundary is computed at each boundary face. This is doneby setting the heat flux for the fluid face equal to the heat flux for the solid face and solving fortemperature. This results in a temperature that satisfies the constraint of an equal and oppositeheat flux for each zonal boundary face. The heat flux uses a second order, one-sided stencil for thetemperature gradient a the boundary face.
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2. The fluid zone is then solved for. Fluxes are computed for each face and boundary conditions areapplied. The unknowns (ie, density, velocity, pressure, turbulence quantities) at each cell center areupdated.
3. The wall temperature is then updated along the zonal boundary due to the updated fluid dynamicsolution. Again the condition of an equal and opposite heat flux is imposed to compute the walltemperature.
4. The solid zone is then solved for. Fluxes are computed for each face and boundary conditions areapplied. The unknowns (temperature) at each cell center are updated.
This completes one iteration cycle. At convergence, a common wall temperature is converged to whichsatisfies the condition that the wall temperature for the fluid is equal to the wall temperature for the solid(Tfw = Tsw), as well as equal and opposite heat transfer fluxes.
IV. Results
The results from the validation cases will now be presented. The first set of cases pertain to a low-speed,turbulent boundary layer with arbitrary wall temperature. Three cases of increasing complexity are studied,namely a heated wall at constant temperature, a step change in wall temperature and a double heat pulse.Numerical results of the Stanton number are compared to experimental results from Reynolds .et. al .1,2
The second set of cases involves supersonic turbulent boundary layers. Again there are three cases inthis set which include an adiabatic wall and a step change in wall temperature with respective ratios of wallto recovery temperature, Tw/Tr, of 1.5 and 2. The final problem is a supersonic cooled axisymmetric nozzle.Both the flow field and the solid wall temperature distributions are computed using GASP’s CHT algorithm.
For both the high- and low-speed cases, the grid is clustered in the y direction using an hyperbolic tangentdistribution such that y+ < 1 for the first cell from the wall. Uniform spacing is used in the x direction andthe grid density is set such that the cell aspect ratio is kept under 1000. At least 40 cells were located insidethe boundary layer. Grid convergence studies were performed to insure that the results are grid independent.For all simulations Roe’s flux difference splitting scheme 20 was used with 3rd order spatial accuracy.
A. Low Velocity Turbulent Boundary Layer with Arbitrary Wall Temperature
Three cases of low velocity turbulent flow over a flat plate are studied. For these cases no significant differencewas found among the turbulence models used. Therefore, only the Wilcox k-ω turbulence model solutionsare shown. A constant value of the turbulent Prandtl number, Prt, of 0.9 was used.
First, a heated wall at constant temperature is considered. The wall temperature is imposed as a boundarycondition. Predicted Stanton numbers (St) are compared against experimental data from Reynolds et al.1,2 Both Menter’s SST and Wilcox’s 1998 k-ω two-equation turbulence models are used as well as theSpalart-Allmaras (S-A) one-equation turbulence model. The free steam properties are
• Te = 300.7K
• Me = 0.111
• ρe = 1.168 kgm3
The temperature of the heated wall is maintained constant at 12.8K above the fluid free steam temperature.Results for St are depicted in figure 2 which closely match experimental data.
The second low-speed case considered is a plate with an isothermal portion followed by an adiabaticregion. The free stream conditions are the following
• Te = 294.7K
• Me = 0.089
• ρe = 1.197 kgm3
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Figure 2: Stanton number for low velocity boundary layer on a heated wall with constant Tw−Te = 12.8Kcompared with experiment from Reynolds et al.1
Figure 3 shows both the experimental wall temperature, used as a boundary condition, and the Stantonnumber. As in the previous case, turbulence modeling doesn’t significantly impact the results. It is importantto note that in the experimental setup, the wall section labeled as adiabatic wasn’t actually insulated; theheaters were simply turned off over that section.2 This explains the negative value of the Stanton numberover that portion as the flow heated over the isothermal region heats up the wall located downstream.
Figure 3: Stanton number for a low velocity boundary layer subjected to wall temperature step comparedwith experiment from Reynolds et al.2
The next case considered is a flat plate with a double pulse in wall temperature with the following freesteam conditions
• Te = 293.3K
• Me = 0.108
• ρe = 1.213 kgm3
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The experimental wall temperature was again used to set the numerical boundary condition. This data alongwith the results for Stanton number are shown in figure 4. As for the previous case, the heater where simplyturned off over the section labeled as adiabatic such that the hot flow heats-up the wall in that region whichexplains the negative Stanton number. Again excellent agreement is seen between the numerical simulationand the experimental results.
(b) Stanton number for a double wall temperature pulse
Figure 4: Stanton number for a low velocity boundary layer subjected to double pulse in wall temperaturecompared with experiment from Reynolds et al. 2
B. Supersonic Turbulent Boundary Layer with a Step in Wall Temperature
The three cases investigated experimentally by Debieve et al. 3 are now presented. To the authors knowledge,these experimental results are the only one for a heated turbulent boundary layer subjected to a step in walltemperature. The Flow conditions and initial boundary layer parameters are
• T0 = 300K ± 5K
• p0 = 0.5× 105Pa± 3%
• Me = 2.3
• δ0 = 10.2mm
• Reθ = 4200
Total temperature and velocity profiles were measured using hot wire anemometry.
1. Generation of Missing Information at the Boundary
When trying to match experimental results using numerical simulations, great care must be taken intoaccurately reproducing the experimental conditions.21,22 For the considered problem this translates intomaking sure that the initial boundary layer profile used for the computation matches the one found inthe wind tunnel. To perform the simulation, the profiles of ρ, u, v, w, p, k, ω must be specified at the inlet.Measurements for these entire variables are rarely available meaning that missing information must begenerated. In our case, the temperature and u-velocity profile are available. Assuming constant pressureacross the boundary layer, the density profile can be computed from the perfect gas law. The consideredflow is two-dimensional such that w = 0. This means that the profiles of the v-velocity, turbulent kineticenergy (k) and specific dissipation rate (ω) must be generated. Different approaches were considered.
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One approach consists in running a flat plate simulation up to the measured integral boundary layer pa-rameter.22 However, since in the experiment, the incoming boundary layer is developed through a converging-diverging nozzle, the computed and measured u-velocity profiles are dissimilar in the outer region of theboundary layer where adjustment is slower. Another approach consist in using the measured u-velocity pro-file to compute the turbulence quantities.21 This was accomplished by using the Cebeci-Smith23 algebraicturbulence model. The approach used is described as follows. For this model, the eddy viscosity is computeddifferently in the outer and inner layer, such that we have
µt =
{µti
, y ≤ ym
µto, y > ym
(5)
where ym is the smallest value of y for which µti= µto
. In the inner layer, µtiis computed as
µti= ρl2mix
[(∂u
∂y
)2
+(
∂v
∂x
)2] 1
2
≈ ρl2mix
∣∣∣∣∂u
∂y
∣∣∣∣ (6)
where lmix is the mixing length given as
lmix = κy[1− e−y+/A+
](7)
whereas in the outer layer we haveµto = αUeδ
∗Fkleb(y; δ) (8)
where Fkleb is the Klebanoff intermittency function given as
The principal Reynolds shear stress is computed from the eddy viscosity
τxy = µt∂u
∂y(12)
Outside the viscous sublayer the kinetic energy is readily computed using the structural parameter a = 0.3
τxy = ak ⇒ k =µt
a
∂u
∂y(13)
whereas the specific dissipation rate ω is given by
ω =ρk
µt(14)
Inside the viscous sublayer we use the asymptotic analysis of the boundary layer performed by Wilcox.8
k = C2yn, y → 0 (15)
with n = 3.23 andω =
C1ν
β∗y, y → 0 (16)
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where β∗ = 9/100 is a closure coefficient of the Wilcox k− ω turbulence model and C1 = 7.20. The value ofC2 is found by matching the expression for k valid outside the viscous sublayer (equation 13) to that validinside (equation 15) at the edge of the viscous sublayer located at y+ = 7. This yields
C2 =(
7µw
ρwuτ
)−n (µt
a
) ∂u
∂y
∣∣∣∣y+=7
(17)
The methodology used to generate k and ω was validated by using a flat plate simulation ran at thesame condition as the experiment from Debieve et al.3 The values of k and ω generated from the u-velocityprofile (obtained from the flat plate simulation) are compared to those obtained directly from the flat platesimulation.
0
0.002
0.004
0.006
0.008
0.01
0.012
0 500 1000 1500 2000 2500K (m2/s2)
y (m
)
Menter SSTK-OmegaGenerated
Figure 5: Generated turbulent kinetic energycompared with computation
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
1.00E+04 1.00E+05 1.00E+06 1.00E+07Omega
y (m
)
Menter SSTK-OmegaGenerated
Figure 6: Generated ω compared with compu-tation
Figure 5 and 6 show a good agreement such that we are confident is this approach. Next the v-velocityprofile must be generated. An approach based on the work of Zhang and Morishita21 is used. The missingv-velocity component is found by integrating the Favre-averaged continuity equation
∂ρu
∂x+
∂ρv
∂y(18)
Which using the fact that the pressure is constant across the boundary layer can be discredited as
vi,j+1 =Ti,j+1
Ti,jvi,j −
Ti,j+1hj
∆x
[ui+1,j
Ti+1,j− ui,j
Ti,j
](19)
Where hj is grid spacing in the y direction and ∆x the spacing between stations i and i + 1. However tocompute v, the u-velocity and temperature profiles must be known at station i and i + 1. To generate thesetwo profile we first look at the integral momentum equation.
dθ
dx=
Cf
2(20)
Which in return requires knowing the skin friction coefficient and the momentum thickness at station i+1.The following analysis is made to get these quantities. Experimental results and analysis have shown thatthe compressible boundary layer follows the law of the wall and the law of the wake when the velocity istransformed according to
uc =∫ (
ρ
ρw
) 12
du =∫ (
Tw
T
) 12
du (21)
Since the pressure is constant across the boundary layer. Starting from the momentum and the energy equa-tion for a two-dimensional boundary layer24 without pressure gradients, a relation between the temperatureand the velocity field can be obtained by acknowledging that the convective terms are negligible close to the
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wall. This simplification enables the integration of the momentum equation which now states that the totalstress is constant across the boundary layer
(µ + µt)∂u
∂y= τw (22)
This result can be substituted into the energy equation which is integrated twice, yielding Crocco’s integral
T = Tw −Prmqwu
cpτw− Prm u2
2cp(23)
where the mixed Prandtl number Prm, defined as
Prm = cpµ + µt
kf + kt(24)
is assumed constant to perform the integration. In practice for a wall-bounded turbulent flow of air Prm ≈Prt ≈ r ≈ 0.9. Using equation 23, equation 21 can be analytically integrated to obtain the Van Driest’stransformation
uc =√
B
[sin−1
(A + u
D
)− sin−1
(A
D
)](25)
where
A =qw
τw(26)
B =2cpTw
prt(27)
D =√
A2 + B (28)
The inverse transformation is given by
u
uτ=
1R
sin(
Ruc
uτ
)−H
[1− cos
(Ruc
uτ
)](29)
where
R =uτ√B
(30)
H =A
uτ(31)
Contrarily to Van Driest I transformation24 the previous is valid for non-unity Pr and Prt as well as non-adiabatic walls since all these effects are included in Crocco’s integral. We use an explicit expression for thelaw-of-the-wall in the inner region and law-of-the-wake in the outer region which is was found by Musker25
uc+ = 5.424tan−1
[2y+ − 8.15
16.7
]+ log10
[(y+ + 10.6)9.6
(y+2 − 8.15y+ + 86)2
]−3.52 + 2.44
{Π
[6
(y
δ
)2
− 4(y
δ
)3]
+(y
δ
)2 (1− y
δ
)}(32)
where Π is the wake parameter26 defined as
Π = 0.55[1− exp
(−0.24
√Reθ − 0.298Reθ
)](33)
We now have all the analytical expression required to compute the velocity and temperature profiles from ei-ther δ or θ. When θ (or δ) and [Tw, Te, Ue, ρe, ρw, yi] are known we can compute δ (or θ) and [u(yi), T (yi), Cf , qw]using the following algorithm first developed by Huang et al.27 and improved by Zhang and Morishita.21
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1. Guess the value of δ (or θ) and uτ . θ = 7/72δ is a reasonable estimate assuming a power law velocityprofile with n = 7 (see Schetz24). uτ can be estimated from Schoenherr’s skin friction correlation24
2. Compute qw using Crocco’s integral (equation 23) evaluated at Ue
3. Compute uce using Van Driest transformation (equation 25) evaluated at Ue
4. Compute Reθ = ρeUeθµw
and evaluate the wake parameter Π using equation 33
5. Compute Reδw= ρwuce δ
µw
6. Compute δ+ = ρwuτ δµw
by solving numerically Reδw= u+
ceδ+ where u+
ceis obtained from equation 32
evaluated at y = δ
7. Compute uτ = δ+µw
ρwδ
8. Compute Cf = 2 Te
Tw
(uτ
Ue
)2
and τw = ρwu2τ
9. Compute u+c at each yi using equation 32
10. Compute uc = u+c uτ and u at each yi from uc using Van driest inverse transformation (equation 32)
11. Compute T at each yi from u using Crocco’s integral (equation 23)
12. Compute a new estimate of δ (or θ) from the u-velocity profile profile using θδ =
∫ 1
0TeuTUe
(1− u
ue
)dη
Step 2 to 12 are repeated until convergence. This algorithm was implemented using Matlab and conver-gence is reached within less than 10 iterations. For higher Mach number, a scaling of y+ was done by Zhangand Morishita.21 This was implemented but wasn’t significant for the Mach number considered herein. Thefollowing steps are used to compute the v-velocity profile.
1. Compute θi from the known temperature and u-velocity profile at station i2. Compute θi+1 for known values of θi and Cf using the momentum integral equation (equation 20)3. From θi+1 compute ui+1,j and Ti+1,j using the previous algorithm4. Compute vi+1,j explicitly for the discretized Favre-averaged continuity equation (equation 19)
The generated v-velocity profile is compared against results from a flat plate simulation at figure 7. Wenotice a difference between the v-velocity profiles among the turbulence models. In the inner layer, thegenerated v-velocity profile agrees with the one obtained with k − ω whereas in the outer layer it reachesthe same free stream value as the one obtained using Spalart-Allmaras turbulence model.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.0 0.5 1.0 1.5 2.0 2.5 3.0v (m/s)
y (m
)
Menter SSTSpalart-AllmarasK-OmegaGenerated
Figure 7: Generated v-velocity component com-pared with computation
0
0.0005
0.001
0.0015
0.002
0.0025
0 0.1 0.2 0.3 0.4 0.5x (m)
Cf
GASPExp
Figure 8: Skin friction coefficient for a supersonicboundary layer with an adiabatic wall comparedwith results from Debieve et al.3
2. Results for the Supersonic Boundary Layer
The first case considered is an adiabatic wall. Very good agreement is found for the skin friction as seenin figure 8. Computed velocity profiles are compared to experimental results in figure 9. At a downstream
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location of 8 cm, good agreement is found close to the wall. The discrepancy away from the wall is probablydue to the effect of pressure gradients in the nozzle. Since the relaxation rate varies proportionally to theinverse of (∂u/∂y), the flow adjusts more quickly close to the wall 3 explaining the better agreement inthe near wall region. At 64 cm, the agreement is better throughout the boundary layer since the flow hasmore time and distance to relax. Overall, better agreement is found when using the experimental velocityprofile as a boundary condition as opposed to a flat plate profile. Menter SST turbulence model gives betteragreement than Wilcox k-ω. Menter SST model and the boundary condition based on the experimentalvelocity profile are used for the two subsequent cases.
0
2
4
6
8
10
12
14
16
18
20
0 100 200 300 400 500 600U (m/s)
y (m
m)
Exp
K-omega IC from flat plate
Menter SST IC from flatplateK-omega IC from Exp
Menter SST IC from Exp
(a) Velocity at 8 cm
0
2
4
6
8
10
12
14
16
18
20
0 100 200 300 400 500 600U (m/s)
y (m
m)
Exp
K-omega IC from flatplateMenter SST IC from flatplateK-omega IC from Exp
Menter SST IC from Exp
(b) Velocity at 46 cm
Figure 9: Velocity profiles for a supersonic boundary layer on an adiabatic wall compared with experimentfrom Debieve et al.3
0
0.0005
0.001
0.0015
0.002
0.0025
0 0.1 0.2 0.3 0.4 0.5x (m)
Cf
GASPExp
(a) Skin friction
0.00E+00
5.00E-04
1.00E-03
1.50E-03
2.00E-03
2.50E-03
3.00E-03
3.50E-03
4.00E-03
4.50E-03
5.00E-03
0 0.1 0.2 0.3 0.4 0.5x (m)
St
GASP Prt=0.9GASP Prt=0.86Exp
(b) Stanton number for Prt equal to 0.9 and 0.86
Figure 10: Skin friction coefficient and Stanton number for a supersonic boundary layer with a step in walltemperature with Tw/Tr = 1.5 compared with results from Debieve .et al.3
For the next two cases, a step change in wall temperature is applied at x = 0 with respective ratios inwall temperature to recovery temperature, Tw/Tr equal to 1.5. and 2. Turbulent Prandtl numbers equalto 0.9 and 0.86 are used. It is observed that such a change doesn’t significantly modify the velocity andtemperature profile. However, a change in turbulent Prandtl number introduces a variation in the Stantonnumber. Very good agreement is found for the skin friction coefficient and Stanton number for both casesas shown in figures 10 and 11 where the computed values are compared to those of Debieve et al.3 Thevariation in Prt shifts the St curve. The two turbulent Prandtl number solutions bracket the experimentaldata.
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0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0.002
0 0.1 0.2 0.3 0.4 0.5x (m)
Cf
GASP
Exp
(a) Skin friction
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0 0.1 0.2 0.3 0.4 0.5x (m)
St
GASP Prt=0.9GASP Prt=0.86Exp
(b) Stanton number for Prt equal to 0.9 and 0.86
Figure 11: Skin friction coefficient and Stanton number for a supersonic boundary layer with a step in walltemperature with Tw/Tr = 2 compared with results from Debieve et al.3
Velocity and temperature profiles are shown in figures 12 and 13. We notice that for both Tw/Tr ratios,the agreement in velocity is better at 8 cm. Agreement in temperature is very good at both stations forboth Tw/Tr ratios. Temperature profiles at both stations closely matches experimental results.
C. Conjugate Heat Transfer in a Cooled Nozzle
Supersonic flow inside a cooled axisymmetric convergent divergent nozzle is investigated. The analysis isbased on the experimental data reported by Back et al.4 Prior to expansion in the nozzle, the air is heated bythe combustion of methanol and directed through a calming section followed by a cooled approach sectionof 18 inch. The mass fraction of the methanol being small (compared to air), the real gas mixture canbe approximated as a perfect gas. The nozzle and the approach section are water-cooled on the outsideof the wind tunnel wall. The temperature distribution inside is experimentally obtained by using threethermocouples embedded at 22 locations along the nozzle wall (the first on the flow side, the second atthe wall center and the third on the cooled side). The uncertainty on the temperature measurements isapproximately 2%.4 The temperature distribution inside the wall is depicted in figure 14 by Back et al.4
reported by Delise and Naraghi.28 The following flow conditions prevail.
• T0 = 843.33 K
• p0 = 5.171× 105 Pa
This case has previously been analyzed by Delise and Naraghi28 and Liu29 et al.. Delise and Naraghi didn’tmodel the heat transfer inside the solid by directly using the wall temperature on the flow side as a boundarycondition. Their analysis is useful as their results show that an algebraic turbulence model can’t be usedto accurately model heat transfer in the vicinity of the throat since the favorable pressure gradient causesa reduction in turbulence intensity which in turn causes a reduction of heat transfer. This phenomenoncan’t be modeled using an algebraic turbulence model (mixing length) or an empirical correlation (N-Rcorrelation). Over predictions of heat transfer of 20% and 70% are obtained by Delise and Naraghi28 for themixing length turbulence model and the N-R correlation respectively.
The wall material wasn’t specified by Back et. al.4 However, the thermal conductivity of the materialkw can be determined from the temperature gradient and the heat flux provided by Back et al.4
q = kw∂T
∂n⇒ kw =
q∂T∂n
≈ q∆n
∆T(34)
where q is the heat flux and kw the thermal conductivity of the wall material. From figure 14, kw is computedby approximating the temperature using the temperature difference between two isotherms separated by a
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Figure 12: Velocity profiles for a supersonic boundary layer with a step in wall temperature at x=0 forratios of Tw/Tr = 1.5 and Tw/Tr = 2 compared with experiment from Debieve et al.3
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Figure 13: Temperature profiles for a supersonic boundary layer with a step in wall temperature at x=0 forratios of Tw/Tr = 1.5 and Tw/Tr = 2 compared with experiment from Debieve et al.3
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Figure 14: Temperature distribution (oF ) inside the nozzle from thermocouple measurements by Back etal.4 reported by Delise and Naraghi28
known distance in a region where all the isotherms are parallel to the wall. In that case the heat conductionis 1-D such that the heat flux is constant across the wall. The following value was obtained
The conjugate heat transfer problem is solved by imposing the temperature at the outside wall takenfrom figure 14. The inside wall temperature doesn’t need to be imposed since the boundary condition forconservation of energy at the interface is used as described in section III. The temperature distributioninside the wall as well as the flow field are being determined. The problem was modeled as axisymmetric.The grid containing 6800 cells is depicted in figure 15.
As for the previous cases, a grid independent solution was achieved by performing a grid refinement study.Four turbulence models were used namely, Spalart-Allmaras, 1998 Wilcox k− ω, Menter’s SST and Chien’sk − ε. Here, turbulence modeling has an impact on wall temperature (and therefore heat transfer rate) aseach model reacts differently to the nozzle favorable pressure gradient which reduces turbulence. The ratioof eddy-viscosity to laminar viscosity is shown at figure 16. Spalart-Allmaras displays a lower initial amountof eddy viscosity and a fast decrease from the favorable pressure gradient starting upstream of the throatwhereas k − ε displays a much higher initial level of eddy-viscosity as well as a significant increase near thethroat. Spalart-Allmaras offers the best agreement with the experimental data as seen at figure 17 where thecomputed internal wall temperature is compared with the experimental results. The nozzle Mach numberand the nozzle wall temperature contours are depicted in figure 18. We notice that the wall temperaturedistribution (figure 18) is similar to the one obtained by Back (figure 14).
V. Conclusions
Convective heat transfer simulations have been performed for both subsonic and supersonic flows andcompared against experimental data. Predicted Stanton numbers for low Mach number turbulent boundarylayers closely agree with experimental results from Reynolds et al. 1,2 for a constant wall temperature,a step and a double pulse in wall temperature. Simulations of supersonic boundary layers with a stepin wall temperature showed good agreement for velocity and temperature profiles when compared with
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(a) Spalart-Allmaras
(b) k − ω
(c) k − ε
(d) Menter’s SST
Figure 16: Ratio of eddy to laminar viscosity for a cooled axisymmetric nozzle
Figure 17: Inside wall temperature for a cooled axisymmetric nozzle compared with experiment from Backet al.4
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(a) Mach number(b) Wall temperature (K)
Figure 18: Mach number and wall temperature contour for a cooled axisymmetric nozzle using Spalart-Allmaras turbulence model
measurements from Debieve et al. 3 as well as for the skin friction and Stanton number. A methodology usedto generate the turbulence information and v-velocity profile based on the work of Zhang and Morishita21 andHuang et al.27 was presented. Using this methodology, better agrement was found between experimental andmeasured velocity and temperature profiles compared to the cases where the input variables were generatedfrom a flat plate simulation. The algorithm enabling the computation of the temperature and velocity profilesfrom one integral boundary layer parameter is particularly useful to generate the inlet velocity profile usedfor a numerical simulation when limited experimental data is available at that location. This shows thetight interconnection between experimental, analytical and numerical methods in modeling complex heattransfer problems. Lastly, the flow field and wall temperature distribution inside a supersonic cooled nozzleis computed using a new CHT algorithm in GASP. Temperature distribution along the inside nozzle wallagreed with measurements from Back et al. 4
Acknowledgment
This work was funded by Arnold Engineering Development Center (AEDC) through the Air Force SBIRproject under contract FA9101-04-C-0035.
References
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3Debieve, J., Dupont, P., Smith, D., and Smits, A., “Supersonic Turbulent Boundary Layer Subjected to Step Changesin Wall Temperature,” AIAA JOURNAL, Vol. 35, No. 1 , Jan 1997.
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Mechanics and Engineering, Vol. 3, 1974, pp. 269–289.12Chien, K.-Y., “Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds Number Turbulence Model,”
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14Neel, R. E., Godfrey, A. G., and Slack, D. C., “Turbulence Model Validation in GASP Version 4,” AIAA Paper 2003-3740,2003, 33rd AIAA Fluid Dynamics Conference and Exhibit.
15Nagano, Y., Closure Strategies for Turbulent and Tansitional Flows, chap. 6 Modelling Heat Transfer in Near-Wall Flows,Cambridge University Press, 2002, pp. 188–247.
16Kays, W., Crawford, M., and Weigand, B., Convective Heat and Mass Transfer Fourth Edition, McGraw-Hill, 2005.17Brinckman, K., Kenzakowski, D., and Dash, S., “Progress in Practical Scalar Fluctuation Modeling for High-Speed
Aeropropulsive Flows,” AIAA 43rd Aerospace Sciences Meeting, Reno. Nevada, 10-13 Jan 2005.18Sommer, T., So, R., and Zhang, H. S., “Near-Wall Variable-Prandtl-Number Turbulence Model for Compressible Flows,”
AIAA Journal, Vol. 31, No. 1 , Jan 1993.19Bradshaw, P., editor, Topics in Applied Physics, Volume 12: Turbulence, Vol. 12, Springer-Verlag, New York, NY, 2nd
ed., 1978.20Roe, P. L., “Approximate Riemann Solvers, Parameter Vectores, and Difference Schemes,” Journal of Computational
Physics, Vol. 43, 1981, pp. 357–372.21Zhang, J. and Morishita, E., “An Efficient and Accurate Way of Posing Inflow Profile Boundary Conditions,” Transacation
of The Japan Society of Aeronautical and Space Science, Aug 2004.22Wong, W. and Qin, N., “A Numerical Study of Transonic Flow in a Wind Tunnel Over 3D bumps,” AIAA 43rd Aerospace
Sciences Meeting, Reno. Nevada, 10-13 Jan 2005.23Smith, A. and Cebeci, T., “Solution of the Boundary-Layer Equations for Incompressible Turbulent Flow,” Proceeding
of the 1968 Heat Transfer and Fluid Mechanics Institute, 1968.24Schetz, J., Boundary Layer Analysis, Prentice Hall, 1993.25Musker, A., “Explicit Expression for the Smooth Wall Velocity Distribution in a Turbulent Boundary Layer,” AIAA
Journal , June 1979.26Cebeci, T. and Smith, A., Analysis of Turbulent Boundary Layers, Academic Press, 1974.27Huang, P., Bradshaw, P., and Coakley, T., “Skin Friction and Velocity Profile Family for Compressible Turbulent Bound-
ary Layers,” AIAA Journal, Vol. 31, No. 9 , Sep 1993.28DeLise, J. and Naraghi, M., “Comparative Studies of Convective Heat Transfer Models for Rocket Engines,” 31st
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