2D Boundary layers with pressure gradients (A)

Underlying Flow Regime 3-03 © copyright ERCOFTAC 2004

Description

Preface

The underlying flow regime (UFR) — 2D Adverse Pressure Gradient Flows — documented here is of major importance in many engineering applications. The behaviour of turbulent flows under adverse pressure gradients is arguably the single most studied phenomena in turbulent engineering flows. It is of major importance e.g. for flows over airfoils and wings, as the separation of the flow determines the safety characteristics of the aircraft. Design tools have to be able to accurately and consistently determine the onset and amount of separation under given flow conditions. For other technical devices, the separation prediction might not be a safety concern, but has a significant impact on the efficiency of the device. Prominent examples are flows through diffusers, flows in turbomachines, flows over cars and other devices imposing an adverse pressure gradient on the boundary layers.

Introduction

The essential physics of the underlying flow regime of a 2D flow under adverse pressure gradients is the deceleration of the near wall flow due to low-momentum fluid near the surface, which cannot sustain the adverse pressure gradient. This leads to a reduction in the wall shear stress and a deviation of the inner scaling from the logarithmic profile. Even though, the flow experiences a deceleration in streamwise direction, the boundary layers are still controlled by the turbulent shear-stress normal to the wall. Anisotropies of the Reynolds stress tensor are not significantly affecting the momentum balance up to separation. Under more severe pressure gradients, the flow experiences flow reversal near the wall, with a region of negative wall shear-stress. Depending on the imposed pressure gradients and the geometry downstream of separation, the flow can reattach onto the surface, or remain separated, leading to a stalled flow topology.

One of the major engineering effects of turbulence is the momentum transfer into the boundary layer, leading to an increased resistance to flow separation. As a consequence, turbulent boundary layers can sustain significantly stronger adverse pressure gradients than their laminar counterparts. This leads to delayed separation and a reduced sensitivity of the boundary layer to pressure gradients. This effect has to be captured accurately by a turbulence model in case of a solution based on the Reynolds averaged Navier-Stokes (RANS) equations.

The following issues related to the flow topology are of significance in the study of adverse pressure gradient flows:

Flows under moderate adverse pressure gradients without separation — these flows experience a reduction of the wall shear-stress and a gradual deviation of the inner scaling of the turbulent boundary layer from the logarithmic profile.

Flows with strong adverse pressure gradients and closed separation bubbles — these flows experience a flow reversal in the near wall region, with a subsequent flow re-attachment.

Massively separated (stalled) flows — these flows separate from the surface without subsequent reattachment. For airfoil and wing flows, this flow regime leads to a dramatic loss of lift and a corresponding increase in drag. For other applications, massive separation results in loss of efficiency and is often accompanied by strong unsteady effects (noise, excitation of the structure, etc.).

Flow recovery — downstream of a reattachment line, the flow accelerates quickly near the wall with a corresponding rise in the wall shear stress.

Most industrial CFD simulations are based on one- or two-equation eddy-viscosity models. The most widely used model is the k-ε model of Jones and Launder (1972). A popular alternative is the k-ω model of Wilcox (1993). It was found in numerous studies that these standard models severely underpredict the sensitivity of the boundary layer to pressure gradients. Particularly the ε-equation based models have shown a strong tendency to underpredict the onset and amount of separation.

For aeronautical applications, optimised eddy-viscosity turbulence models have been developed with improved separation prediction capabilities (Menter 1994, Spalart Almaras 1994) in the late 80s and early 90s. These modern turbulence models offer a level of accuracy for the prediction of the separation line, which is sufficient for most engineering applications. This judgement is supported by numerous validation studies, which are however mostly limited to two-dimensional and axisymmetric flows. For three-dimensional flows, the evidence is much weaker, mainly due to a lack of experimental data and the increased validation effort. In addition, progress has been made with more complex formulations based on Explicit Algebraic Reynolds Stress Models (EARSM) (Gatski and Speziale 1993, Wallin and Johannson 2002) or Second Moment Closure (SMC) (Launder et al. 1975, Speziale et al. 1991, Wilcox 1993) models to properly predict the separation onset.

Despite the improvements in the prediction of the separation line, all RANS models show a consistent deviation between experimental data and numerical predictions in the region downstream of a separation zone. Typically it is found that models, which predict the correct separation onset, have a tendency to overpredict the extent of the separation zone (Johnson et al. 1994). Furthermore, the models have a tendency to predict a delayed flow recovery downstream of reattachment. The improvement of turbulence models in this area is an important next step in the development of accurate turbulence models for wall-bounded flows. This is currently an active research area.

Figure 1 shows a typical comparison of velocity profiles for an asymmetric diffuser (Obi et al. 1993). The picture is not intended for a detailed discussion, but to show the principal behaviour of current turbulence models using the k-ε and the SST model as an example. The k-ε model clearly fails to predict the separation observed in the experiment. It therefore misses the main characteristics of this flow. The SST model avoids this deficiency and gives a much more accurate representation of the velocity profiles in the separation zone. On closer inspection, the model shows a slower flow-recovery downstream of reattachment than observed in the experiment. It was found in numerous studies that all RANS models, which give the proper onset of separation, predict a reduced flow recovery, and in more severe cases a prolonged separation zone (for very detailed turbulence model validation studies, see ERCOFTAC , 1999, 2002).

Figure 1 Velocity profiles for asymmetric diffuser. Comparison of the k-ε and SST model with experimental data.

Review of UFR studies and choice of test case

Historically, turbulent flows with adverse pressure gradients were the most widely studied flows in turbulence modelling. There is a large number of testcases with detailed experimental data covering a wide range of pressure gradients and flow regimes. Earlier experiments (Clauser 1954, Schubauer and Spangenberg 1960, Perry 1966, Samuel and Joubert 1974)) typically investigated flows under adverse pressure gradients without separation. This was motivated by the use of boundary layer codes, which could not march into the separation zone. More recent experiments cover flows without and with separation and offer more detailed experimental data due to the improvements in measurement techniques (Perry and Schofield 1973, Simpson et al. 1981, Schofield 1981, Bogar et al. 1983, Bachalo and Johnson 1985, Buice and Eaton 1997). In addition to the classical data on velocity profiles, the newer data sets provide also the turbulent stresses and more accurate wall shear-stress data. For a review on experiments see Hancock (2000). Recently, the experimental database has been augmented by results of Direct Numerical Simulations (DNS) for turbulent separated flows at low Reynolds numbers. (Na and Moin 1998, Spalart and Coleman 1997).

Over the years a large number of validation studies have been carried out by individual groups as well as in the framework of validation workshops (e.g. ERCOFTAC/IAHR/COST workshops). As part of two important workshops in Stanford (Kline et al. 1969, Kline et al. 1981) numerous boundary layers under adverse pressure gradients have been studied.

The CS0 diffuser testcase by D. Driver (1991) has been selected for this UFR, because it provides a valuable data-set for turbulence model development and validation. One of its main advantages is that the flow geometry is axisymmetric, avoiding the influence of the tunnel side-walls, which often negatively affects the comparison in 2D geometries. Furthermore, the experiment offers an extensive set of experimental data, ranging from the wall pressure and wall shear-stress distribution to velocity profiles and turbulent stresses. The CS0 case has also proven to be a highly consistent data set, where the data from different measurement techniques do not result in contradictions in the interpretation of the flowfield.

The CS0 testcase is highly sensitive to the capability of a model to accurately predict separation. Due to the gradual increase in the pressure gradient, the separation point varies strongly between turbulence models. This is not always the case for diffuser flows with a sharp geometric change in cross section (Obi et al. 1993), which are less discriminating between the models. On the other hand, the CS0 case has a relatively small separation zone and does not extend far downstream of the separation bubble. It is therefore not a suitable testcase for studies on flow recovery.

Contributors: Florian Menter - AEA Technology