# UFR 3-09 Test Case

## Contents

# Impinging jet

Underlying Flow Regime 3-09 © copyright ERCOFTAC 2004

# Test Case

## Brief description of the study test case

The flow consist of an air jet issuing from a circular pipe at ambient temperature. The flow impinges orthogonally on a flat plate heated from below with a constant heat flux.

After proper normalization by the inlet bulk velocity, the pipe diameter and the wall heat flux, the parameters defining the flow are:

Reynolds number Re=Ub D / nu = 23,000

Prandtl number of air Pr=0.71

Nozzle-to-plate distance:diameter ratio H/D=2

The flow is axisymmetric. The geometry is defined in Fig. 1.

The mean velocity magnitude, the Reynolds stresses u'u', v'v' and u'v', as well as the Nusselt number distribution on the flat plate are available.

The quantity that allow judgment of the success or failure of a CFD calculation is mainly the Nusselt number distribution, in particular from an industrial point of view. As regards the turbulence models assessment, many other aspects are critical: prediction of the mean velocity in the developing radial wall jet; prediction of the turbulent energy in the impingement region; prediction of the anisotropy in the vicinity of the impingement point, and in particular, reproduction of the damping of the intense fluctuations normal to the wall by the blocking effect of the wall.

## Test Case Experiments

4.a Baughn and Shimizu temperature measurements

The uniform heat flux is established by electrically heating a very thin vacuum deposited gold coating on a plastic substrate. The surface temperature distribution is measured using liquid crystals.

The length:diameter ratio of the pipe was 72:1, which ensures with a high degree of confidence that the pipe flow is fully developed when it reaches the pipe exit. As stated before, this is very convenient for defining all the inlet quantities from a separate pipe flow computation. The turbulence intensity at the pipe exit is about 4 percent. 4 different nozzle-to-plate distances are used (H/D=2; 6; 10; 14). The Reynolds number is 23,750.

Accuracy:

- The axisymmetry of the flow is considered as good since the color band of the liquid crystals was almost a perfect circle.

- A radiation correction, using the measured emissivity of 0.5, was applied tyo determine the convective component of the surface heat flux. The radiation correction is less than 5 percent of the total.

- The conduction losses through the substrate were less than 1 percent.

- The temperature resolution of the green color liquid crystal is better than 0.1° C, while the temperature differences are of the order of 10° C.

- A standard uncertainty analysis was performed. The uncertainty in the Nusselt number was estimated at 2.4 percent, at 2.3 percent in the Reynolds number and at less than 1 percent in r/D and z/D.

4.b Cooper et al. flow field experiment

The experiments were carried out in air (Pr=0.71) with a copper pipe of 26 mm internal diameter. Note that a second series of measurements were performed with a brass pipe of 101.6 mm internal diameter which confirm the results of the first experiments.

The flow impinges on a rectangular test plate measuring 1275 mm x 975 mm, made from printed-circuit board fixed to a 25 mm thick plywood backing.

A two-channel hot-wire anemometer allows the measurement of the magnitude of the mean velocity in the plane x-r, and of the Reynolds stresses uu, vv and uv.

Two different Reynolds numbers were used in the experiments and five different values of the nozzle-to-plate distance parameter H/D. However, for different reasons, the case that received the most attention for validating turbulence models and CFD codes is the case at Re=23,000 and H/D=2. Therefore, this documentation is focused on this particular case.

The length:diameter ratio of the pipe was 80:1, which is fully sufficient for ensuring that the flow in the pipe is fully developed.

The ratio of the dimensions of the rectangular plate to the pipe diameter are 49 and 37.5, which eliminates the possibility of end effects.

Accuracy :

- In order to check the axisymmetry of the flow, profile were measured at 90° intervals. The profiles were indistinguishable from one another. This also supports the absence of end effects due to the plate finiteness (since the plate is not a square, if end effect were present, the profiles at 90° intervals would differ).

- The hot wire probe was aligned with an error less than 0.3°.

- The calibration error is less than 2%.

- Therefore, the author evaluate the absolute accuracy of their data to: 2% for the maximum mean velocities; 4% for u' and 6% for v' (root-mean-square fluctuating velocities); 9% for uv.

**Summary:**

Though the temperature and flow fields measurements are from two different experiments, they are fully consistent, so they can be considered as a single experiment.

Inlet conditions: the flow is a fully developed pipe flow, at least a few diameters before the pipe exit. The air issuing from the pipe is at ambient temperature.

Wall conditions: the plate is smooth. A constant heat flux is imposed at the wall.

Outlet conditions: the test section is 3/4 open (only limited by the plate).

Parameters of the flow: Re=23,000; Pr=0.71; H/D=2.

The database is available from the ERCOFTAC Classic collection, at this link.

As shown in Fig. 1, single-wire measurements (mean velocity and uu) are available at 7 different r/D locations: 0, 0.5, 1, 1.5, 2, 2.5, 3.

They are provided in the files:

ij2lr-??-sw-mu.dat (mean velocity)

ij2lr-??-sw-uu.dat (Reynolds stress uu)

where ?? stands for the location.

Cross-wire measurements (vv and uv) are also available at 4 locations among the 7: 0.5, 10, 2.5, 3.

They are provided in the files:

ij2lr-??-cw-vv.dat

ij2lr-??-cw-uv.dat

The Nusselt number distribution on the plate is also provided in the file ij2lr-nuss.dat.

References:

Baughn, J. & Shimizu, S. 1989, Heat transfer measurements from a surface with uniform heat flux and an impinging jet. J. Heat Transfer 111, 1096-1098.

Cooper, D., Jackson, D. C., Launder, B. E. & Liao, G. X. 1993, Impinging jet studies for turbulence model assessment-I. Flow-field experiments. Int. J. Heat Mass Transfer 36, 10-10.

## CFD Methods

All the CFD simulations presented below are RANS simulations performed at Re=23,000, H/D=2 and Pr=0.71, on a 2D grid (axisymmetry).

Label |
Authors |
year |
Turbulence model |
Thermal flux model |
Wall treatment |
Mesh |

a |
Craft et al. |
1993 |
k-epsilon Launder-Sharma |
Prt=0.9 |
Integ |
70x80 |

b |
RSM Gibson-Launder |
Linear Heat Flux TM |
k-epsilon Launder-Sharma |
70x80 | ||

c |
RSM New wall-refl. |
Linear Heat Flux TM |
k-epsilon Launder-Sharma |
70x80 | ||

d |
RSM Cubic. |
NL Heat Flux TM+eps-th TM |
k-epsilon Launder-Sharma |
70x80 | ||

e |
Dianat et al. |
1996 |
k-epsilon |
No heat transfer |
WF |
102x93 |

f |
RSM Jones&Musonge |
No heat transfer |
WF |
102x93 | ||

g |
Gibson & Harper |
1997 |
k-epsilon Launder-Sharma |
Prt=0.91 |
Integ |
112x186 |

h |
q-dzeta |
prt=0.91 |
Integ |
112x186 | ||

i |
q-dzeta |
q_theta-dzeta_theta |
Integ |
112x186 | ||

j |
q-dzeta+Yap |
q_theta-dzeta_theta |
Integ |
112x186 | ||

k |
Behnia et al. |
1998 |
k-epsilon Launder-Sharma |
Prt=K&C |
Integ |
120x120 |

l |
V2F |
Prt=0.73, 0.85, 0.92, K&C |
Integ |
120x120 | ||

m |
Park & Sung |
2001 |
k-epsilon |
Prt=0.9 |
WF |
141x93 |

n |
V2F |
Prt=0.9 |
Integ |
|||

o |
k-epsilon-fmu |
Prt=0.9 |
Integ |
|||

p |
k-epsilon-fmu New |
Prt=0.8, Prt=0.9, Prt=K&C |
Integ |
|||

q |
Thielen et al. |
2001 |
k-epsilon |
Prt=cnst |
WF |
72x100 |

r |
RSM LRR |
No heat transfer |
WF |
72x100 | ||

s |
V2F |
Prt=cnst |
Integ |
100x100 | ||

t |
RSM EBM |
GGDH |
Integ |
100x100 | ||

u |
Manceau et al. |
2002 |
V2F |
Prt=0.9 |
Integ |
150x120 |

v |
Rescaled V2F |
Prt=0.9 |
Integ |
150x120 | ||

w |
Esch et al. |
2003 |
k-epsilon |
Prt=0.9 |
WF |
450x300 |

x |
SST |
Prt=0.9 |
Integ |
450x300 |

The lettering of the following descriptions corresponds to Table 1.

**Computation a:**

- Code: TEAM, developed at UMIST.

- Equations solved:

Mean momentum transport equations using an eddy viscosity hypothesis for the Reynolds stress.

k and epsilon transport equations: Launder-Sharma model with Yap correction.

Heat transport equation using an eddy diffusivity hypothesis for the turbulent fluxes, with a constant turbulent Prandtl number (Prt=0.9).

Launder, B. E. & Sharma, B. I. 1974, Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc. Letters in Heat and Mass Transfer 1, 131-138.

- Solution algorithm: SIMPLE

- Numerical discretization: third-order QUICK scheme. Steady state computation.

- Computational domain: 0.5 diameters above the pipe exit; radial distance of six diameters.

- Grid: 70 (radial) x 80 (axial) nonuniform grid.

No information given about the stretching and the near-wall cell size.

- Boundary conditions

Inlet (x=2.5 D): All profiles prescribed via separate parabolic computation using Gibson-Launder RSM. Ambient temperature.

Walls: All variables (except for temperature) set to zero including pseudo-dissipation. Uniform heat flux imposed on the plate.

Right boundary (r=6 D): uniform static pressure. The constraint on the turbulent variables depend on whether the flow is entering or leaving the domain: In-flowing fluid is assigned zero values of turbulent stress and dissipation rate while, for fluid leaving, zero gradient conditions are applied.

Remaining upper boundary (x=2.5 D): same as right boundary.

Center line (r=0): symmetry conditions.

No test about the influence of the boundary conditions.

- Numerical accuracy

No information about convergence criteria.

A grid sensitivity analysis was performed in Craft's thesis, showing that the grid used here is sufficient:

Craft, T. J. 1991, Second moment modelling of turbulent scalar transport. Ph.D. Thesis, Univ. Manchester Inst. Science and Techn.

- Quantities given in the paper

Profiles of u' (rms fluctuating velocity), v', uv and velocity magnitude at different locations: r/D=0.5, 1.0, 2.50. Kinetic energy profiles in the near-wall region. Nusselt number distribution on the plate.

**Computation** b:

Same as computation a, except for:

- Equation solved

Mean momentum transport equations

Reynolds stress transport equations: Gibson-Launder model, i.e., linear pressure-strain model and wall echo terms.

Epsilon transport equation.

Heat transport equation.

Turbulent heat flux transport equations: linear model.

Temperature variance transport equation, with a constant time-scale ratio for the temperature variance dissipation rate.

Gibson, M. M. & Launder, B. E. 1978, Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech. 86, 491-511.

- Near-wall treatment: in the near-wall region, the Launder-Sharma low-Reynolds number model (see computation a) is integrated down to the wall, with the Yap correction in the epsilon equation. The zone in which the Launder-Sharma model is used is located between the plate and a radial line at a distance from the wall such that the average turbulent Reynolds number is about 150.

**Computation** c:

Same as computation b, except for:

Reynolds stress transport equations: a new wall echo term is used.

**Computation** d:

Same as computation b, except for:

Reynolds stress transport equation: Craft et al. cubic pressure-strain model

Turbulent heat flux transport equations: a nonlinear formulation is used.

Temperature variance transport equation: no hypothesis on the time-scale ratio; a transport equation is also solved for the temperature variance dissipation rate.

Craft, T. J., Launder, B. E., Tselepidakis, D. P. 1989, 7th Symp. Turb. Shear Flows, Stanford, California, USA.

**Computation** e:

- Code: home made code.

- Equations solved:

Mean momentum transport equations using an eddy viscosity hypothesis for the Reynolds stress.

k and epsilon transport equations: standard k-epsilon model.

- Numerical discretization: second-order TVD scheme. Steady state computation.

- Computational domain: no information.

- Grid: 102 (radial) x 93 (axial) nonuniform grid. Expansion ratio is less than 1.05.

- Boundary conditions

Inlet: All profiles prescribed via separate parabolic computation using the same model. Ambient temperature.

Walls: wall functions.

Right boundary: uniform pressure.

Remaining upper boundary: same as right boundary.

Center line (r=0): symmetry conditions.

The influence of the location of the upper and right boundary conditions was checked. The location used in the present calculation is then chosen sufficiently far.

- Numerical accuracy

No information about convergence criteria.

A grid sensitivity analysis was performed, showing the independence of the solution on the grid: the results obtained with the finer grid are presented.

- Quantities given in the paper

Profiles of u', v', uv and velocity magnitude at different locations: r/D=0.5, 1.0, 2.50, 3. u' is also given on the stagnation line (r=0).

**Computation** f:

Same as computation e, except for:

Equations solved:

Mean momentum transport equations.

Reynolds stress transport equations: Jones-Musonge linear model.

Epsilon transport equation.

**Computation** g:

- Code: home made code.

- Equation solved:

Same as computation a, but without Yap correction.

- Solution algorithm: SIMPLE.

- Numerical discretization: no information.

- Computational domain: 10D in radial direction. 2.5D in axial direction, except for the inlet boundary, located at the pipe exit (2D).

- Grid: 186 (radial) x 112 (axial) nonuniform grid. First near-wall nodes are located around y+=1.

- Boundary conditions

Inlet (x=2D): All profiles prescribed via separate parabolic computation using the model of computation h. Ambient temperature.

Walls: all variables (except for temperature) are set to zero at the wall, including pseudo-dissipation. Constant heat flux on the plate.

Right boundary (r=10 D): uniform static pressure. The constraint on the turbulent variables depend on whether the flow is entering or leaving the domain: In-flowing fluid is assigned zero values of turbulent quantities while, for fluid leaving, zero gradient conditions are applied.

Remaining upper boundary (x=2.5 D): same as right boundary.

Center line (r=0): symmetry conditions.

- Numerical accuracy

No information about convergence criteria.

A grid sensitivity analysis was performed, showing the independence of the solution on the grid: the results obtained with the finer grid are presented.

- Quantities given in the paper

Profiles of uv and velocity magnitude at different locations: r/D=0.5, 1.0, 2.50, 3. Development of the radial wall jet.

**Computation** h:

Same as computation g, except for:

Mean momentum transport equations using an eddy viscosity hypothesis for the Reynolds stress.

Transport equation for q=k^0.5 and its dissipation rate dzeta.

Heat transport equation using an eddy diffusivity hypothesis, with a constant turbulent Prandtl number (Prt=0.91).

**Computation** i:

Same as computation g, except for:

- Equations solved:

Mean momentum transport equations using an eddy viscosity hypothesis for the Reynolds stress.

Transport equation for q=k^0.5 and its dissipation rate dzeta.

Heat transport equation using an eddy diffusivity hypothesis.

Transport equation for q_theta=theta^2^0.5 and its dissipation rate dzeta_theta.

**Computation** j:

Same as computation i, except for:

The Yap correction term is added to dzeta transport equation.

**Computation** k:

- Code: home made finite difference code.

- Equation solved:

Mean momentum transport equations using an eddy viscosity hypothesis for the Reynolds stress.

k and epsilon transport equations: Slightly modified Launder-Sharma model (no Yap correction) (see computation a).

Heat transport equation using an eddy diffusivity hypothesis for the turbulent fluxes, with a turbulent Prandtl number given by the Kays-Crawford correlation.

Kays, W. M. & Crawford, M. E. 1993. Convective heat and mass transfer. Third Edition, Mc Graw-Hill

- Numerical discretization: third order upwind biased scheme for convective terms.

- Computational domain: 10 D in radial direction. 4 D in axial direction.

- Grid: 120 (radial) x 120 (axial) nonuniform grid. First near-wall nodes are located around y+=1.

- Boundary conditions

Inlet (x=4 D): All profiles prescribed via separate computation using the same model. Ambient temperature.

Walls: All variables (except for temperature) are set to zero, including pseudo-dissipation rate. Constant heat flux on the plate.

Right boundary (r=10 D): uniform static pressure.

Remaining upper boundary (x=4 D): same as right boundary.

Center line (r=0): symmetry conditions.

- Numerical accuracy

A grid sensitivity analysis was performed, showing the independence of the solution on the grid..

- Quantities given in the paper

Profiles of mean velocity magnitude at different locations: r/D=0, 0.5, 1.0, 2.5. Nusselt number distribution.

**Computation** l:

Same as computation k, except for:

- Equations solved:

Mean momentum transport equations.

k and epsilon transport equations.

Wall blocked turbulent energy scale v2.

Elliptic relaxation equation for f, the source term of v2.

Heat transport equation: a turbulent diffusivity hypothesis is used. Tests are carried out to investigate the influence of the value of the turbulent Prandtl number: Prt=0.73, Prt=0.85, Prt=0.92, Prt=Kays-Crawford correlation (see computation k). The influence is weak, but the Kays-Crawford correation yields the best results: therefore, it is used in the following of the study.

-Boundary conditions:

At the walls, epsilon and f are set to value depending on k and v2, respectively.

**Computation** m:

- Code: home made finite difference code.

- Equation solved:

Same as computation a.

- Solution algorithm: SIMPLEC.

- Numerical discretization: second order HLPA scheme.

- Computational domain: 10 D in radial direction. 5 D in axial direction, except at the inlet boundary, located at the pipe exit (2 D).

- Grid: 141 (radial) x 93 (axial) nonuniform grid.

- Boundary conditions

Inlet (x=2 D): All profiles prescribed via separate computation using the same model. Ambient temperature.

Walls: All variables (except for temperature) are set to zero, including pseudo-dissipation rate. Constant heat flux on the plate.

Right boundary (r=10 D): entrainment condition. The constraint on the turbulent variables depend on whether the flow is entering or leaving the domain: In-flowing fluid is assigned zero values of turbulent quantities while, for fluid leaving, zero gradient conditions are applied.

Remaining upper boundary (x=4 D): same as right boundary.

Center line (r=0): symmetry conditions.

- Numerical accuracy

A grid sensitivity analysis was performed. The results presented are obtained with the finest grid.

- Quantities given in the paper

Profiles of mean velocity magnitude at different locations: r/D=0, 1.0, 2.5. Nusselt number distribution. Profiles of k and eddy viscosity on the stagnation line.

**Computation** n:

Same as computation m, except for:

- Equations solved: V2F model (see computation l).

Turbulent Prandtl number Prt=0.9.

- Boundary conditions at the walls: (see computation l)

**Computation** o:

Same as computation m, except for:

-Equations solved:

Mean momentum transport equations with an eddy viscosity hypothesis.

k and epsilon transport equations.

Elliptic relaxation for the function f_w which allows the damping of the eddy viscosity.

Heat transport equation with an eddy diffusivity hypothesis, with a constant turbulent Prandtl number Prt=0.9.

- Boundary conditions at the walls: epsilon is set to a non-zero value depending on k.

**Computation** p:

Same as computation o, except for:

-Equations solved:

The eddy viscosity formulation is modified to allow the accounting of non-equilibrium effects far from the wall.

Tests are performed to evaluate the influence of the value of the turbulent Prandtl number: Prt=0.8, Prt=0.9 or Prt=Kays-Crawford correlation. Only marginal discrepancies are noted: Prt=0.9 is used in the remaining of the study and with other models (computations m, n, o).

**Computation** q:

- Code: home made finite volume code.

- Equation solved:

Standard k-epsilon model (see computation e).

Heat transport equation with a turbulent diffusivity hypothesis, with a constant turbulent Prandtl number.

- Numerical discretization: second order scheme.

- Computational domain: 8 D in radial direction. 2.5 D in axial direction.

- Grid: 100 (radial) x 72 (axial) nonuniform grid.

- Boundary conditions

Inlet (x=2.5 D): All profiles prescribed via separate developed pipe flow computation. Ambient temperature.

Walls: wall functions. Constant heat flux on the plate.

Right boundary (r=8 D): outlet boundary condition.

Remaining upper boundary (x=4 D): constant pressure boundary condition.

Center line (r=0): symmetry conditions.

- Numerical accuracy

No information.

- Quantities given in the paper

Profiles of mean velocity magnitude, uu, vv and uv at different locations: r/D=0.5, 1.0, 2.5, 3.0. Nusselt number distribution. Profiles u' (rms value of the wall normal fluctuation) on the stagnation line.

**Computation** r:

Same as computation q, except for:

- Equations solved:

Mean momentum transport equations.

Reynolds stress transport equations, with linear Gibson-Launder pressure-strain model (see computation b).

Epsilon equation.

- Quantities given in the paper: no Nusselt number distribution.

**Computation** s:

Same as computation q, except for:

- Equations solved:

Mean momentum transport equations.

V2F turbulence model (see computation l).

Heat transport equation with a turbulent diffusivity hypothesis, with a constant turbulent Prandtl number.

- Grid: 100 (radial) x 100 (axial) nonuniform grid. Size of the near-wall cells: about y+=3.

- Boundary conditions at the walls: mean velocities, k and v2 are set to zero at the wall. Epsilon and f are set to non-zero values depending on k and v2, respectively. A constant heat flux is imposed on the plate.

**Computation** t:

Same as computation s, except for:

- Equations solved:

Mean momentum transport equations.

Reynolds stress transport equations, using the elliptic blending model for the pressure-strain and dissipation terms.

Manceau, R. & Hanjalic, K. 2002. Elliptic Blending Model: A New Near-Wall Reynolds-Stress Turbulence Closure. Phys. Fluids 14, 744-754.

Epsilon equation.

Elliptic relaxation equation for the blending factor alpha.

Heat transport equation, using a generalized gradient diffusion hypothesis for the turbulent heat fluxes.

- Boundary conditions at the walls:

Mean velocities, Reynolds stress and alpha are set to zero at the walls. Epsilon is set to a non-zero value depending on the turbulent energy. A constant heat flux is imposed at the wall.

**Computation** u:

- Code: commercial software Star-CD.

- Equation solved:

Same as computation n

- Numerical discretization: second order scheme (central differencing for the convective terms).

- Computational domain: 10 D in radial direction. 4 D in axial direction.

- Grid: 150 (radial) x 120 (axial) nonuniform grid. The first near-wall node is located below y+=1.

- Boundary conditions

Inlet (x=4 D): All profiles prescribed via separate developed pipe flow computation with the same model. Ambient temperature.

Walls: same as computation n.

Right boundary (r=10 D): outlet boundary condition.

Remaining upper boundary (x=4 D): symmetry boundary condition.

Center line (r=0): symmetry conditions.

- Numerical accuracy

No information.

- Quantities given in the paper

Profiles of mean velocity magnitude at different locations: r/D=0.0, 0.5, 1.0, 1.5, 2., 2.5, 3.0. Nusselt number distribution.** **

**Computation** v:

Same as computation u, except for:

-Equations solved:

The Manceau et al. rescaled V2F model is used instead of the V2F model.

Manceau, R., Carlson, J. R. & Gatski, T. B. 2002. A rescaled elliptic relaxation approach: neutralizing the effect on the log_layer. Phys. Fluids, 14-11, 3868-3879.

**Computation** w:

- Code: commercial software CFX

- Equation solved:

Same as computation q with Prt=0.9

- Numerical discretization: finite volume. No details given about the order.

- Computational domain: 13D in both directions;

- Grid: 450 (radial) x 300 (axial) nonuniform grid. The mean location of the near-wall node is y+=2.1.

- Boundary conditions

Inlet (x=13 D): All profiles prescribed via separate developed pipe flow computation with the same model. Ambient temperature.

Walls: Scalable wall function.

Right boundary (r=13 D): constant pressure.

Remaining upper boundary (x=13 D): constant pressure.

Center line (r=0): symmetry conditions.

- Numerical accuracy

Grid refinement analysis performed.

- Quantities given in the paper

Profiles of mean velocity magnitude at different locations: r/D=0.5, 1.0, 2.5.

Nusselt number distribution.

**Computation** x:

Same as computation w except for :

- Equation solved:

k-omega SST model (integration down to the wall).

Menter, F.R., 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J., 32-8, 1598-1605.

(except for a slight modification explained in the paper)

© copyright ERCOFTAC 2004

Contributors: Remi Manceau - Université de Poitiers