The plane wall jet

Best Practice Advice for the UFR

Key Physics

The test case is a two-dimensional plane wall jet in infinite surroundings. It can formally be defined as a shear flow directed along a wall where, by virtue of the initially supplied momentum, at any station, the streamwise velocity over some region within the shear flow exceeds that in the external stream, Launder and Rodi (1981). The plane turbulent wall jet is a fairly complex test case, since there is interaction between the inner layer of the flow (from y = 0 to y = ym, where U = Um) with small turbulence scales and the outer layer (y > ym) with larger scales. The inner flow resembles a turbulent boundary layer, and the outer part of the flow is similar to a free jet. There also exists a region of negative production of turbulent kinetic energy, between the point of zero shear stress (uv = 0) close to the wall and the point dU/dy = 0 further out.

In spite of the fact that the test case has been the subject of two workshops, the predictions are not entirely satisfactory. Based on an assessment of the existing computations and the comparisons with the experimental data, and the present state-of-the-art of CFD and turbulence modelling, we will here try to formulate some best practice advice related to this test case or underlying flow regime. (It should be noted, of course, that such advice need to be updated at regular intervals.)

Numerical issues

Numerical diffusivity should be minimized by using a suitable higher-order numerical scheme. This is important not only in the transport equations governing the mean flow field, but also for the turbulence equations. Upwind schemes should not be employed.

No complete study of grid independence has been made for this test case (largest grid studied was 326*217 nodes). It can be estimated that at least 600 *400 (= 240 000) nodes are necessary when using wall functions. A fully resolved near-wall computation requires the first grid point to be located at y+ < 1, with at least 20 points, say 25, for y+ < 50. With full resolution down to the wall (two vertical and two horizontal walls), the grid requirement increases to slightly less than 300 000 cells. With the sharp increase in cheap computer capacity made available by means of PC clusters, it should not be a problem to create a sufficiently fine grid to produce a grid-independent solution for this two-dimensional problem. In fact, it is fully possible to perform a formal study of grid independence by using three different grids, in accordance with the ERCOFTAC Best Practice Guidelines for Industrial CFD (Casey & Wintergerste, 2000).

Computational domain and boundary conditions

It is recommended that the flow field in the whole domain is computed. Then the boundary conditions are also easily specified: only one inlet boundary (the slot with prescribed mean velocity profile at x = 0) and one outlet boundary with prescribed flow rate. Solid walls at the bottom (y = 0) and at the end walls (x = 0.0 m and 7.0 m) with zero slip, and an upper water surface at y = 1.45 m with zero friction. The flow is assumed two-dimensional and stationary.

Since the flow field at the inlet is not completely determined (only the streamwise mean velocity and turbulence intensity were measured), some assumptions must be made regarding the unknown quantities v' and w'. These quantities can be estimated from knowledge of the turbulence intensities in turbulent boundary layers. Alternatively, one can put corresponding to isotropic turbulence. An estimate of the turbulence dissipation rate ε or a turbulent length scale L is also needed. With a slot height of 9.6 mm and initial boundary-layer thickness at x = 0 of about 1.5 mm, a suitable length scale is of the order 0.5 - 3 mm. Limited sensitivity studies were performed and presented at the workshops, but no clear conclusion was reached.

Physical modelling

• The growth rate and the skin friction of the wall jet are generally overpredicted in computations using the standard k-ε model with wall functions (+25-30 %). The use of wall functions to calculate plane wall jets is not appropriate, and the near-wall region needs to be resolved.
• However, near-wall two-equation models which can predict plane free jets and the near-wall region of wall-bounded simple flows can also give reasonable results on skin friction and growth rate for turbulent plane wall jets.
• No eddy-viscosity model can predict the region of negative production of turbulent kinetic energy.
• Some of the RSM models seem to be able to predict the skin friction quite well (+4 %) and the growth rate reasonably well (+13 %). Also here the near-wall region needs to be resolved.

Application uncertainties

There should be no significant application uncertainties associated with the inlet boundary conditions, except for the length scale. Here some further sensitivity tests are necessary. The boundary conditions on all other boundaries are well specified for the present test case.

Recommendations for further work

In view of recent advances in computational power and turbulence modelling, the appearance of the ERCOFTAC Best Practice Guidelines for Industrial CFD, and with more complete experimental data available today, it appears timely to reconsider this test case. Now the computer capacity will permit to obtain a grid-independent solution of the wall-jet test case.

Wall functions should be abandoned and replaced by a physically more appealing near-wall model formulation. To this end, the elliptic relaxation approach originally proposed by Durbin for a linear eddy-viscosity model (Durbin, 1991), and later for a full RSM model (Durbin, 1993), constitutes a most attractive candidate. Pettersson Reif (2000) extended this approach to also include nonlinear eddy-viscosity closures.

Further improvements based on eddy-viscosity models, including EARSM:s, are doubtful since an instantaneous equilibrium between stresses and strains is inherently assumed. These closures therefore do not have the ability to predict the negative turbulence production region that originates from the offset position of uv=0 and dU/dy=0. This region most certainly affects the growth rate of the wall jet.

This intricate feature of the turbulent wall jet can only be captured with a full RSM computation.