Sicheng LI, Chenyuan BAI, Jing LIN, Ziniu WU

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

KEYWORDS

Abstract Wagner problem is originally concerned with inviscid flow and unsteady force due to a small step motion, or attaining of a small angle of attack, of an airfoil in an initially uniform flow and has been studied recently for inviscid flow with large amplitude step motion. Here we propose to consider turbulent Wagner problem for a plate that is initially covered with a mixed laminarboundary layer on both sides and is set into step motion of small or large amplitude and in direction normal to the plate. The evolution of skin friction and transition region in time are examined numerically. It is found that transition region unexpectedly changes direction of movement for small amplitude of step motion while global transition or laminarization exists for large amplitude step motion. The significance of this study is twofold. First, the present study treated a new and interesting problem since it combines two problems of fundamental interests, one is Wagner problem and the other is boundary layer transition.Second, the present study appears to show that the pressure gradient normal to the airfoil and caused by discontinuous step motion may have subtle influence on transition and the mechanism of this influence deserves further studies.

1. Definition of turbulent Wagner problem with transition

The problem we consider is schematically displayed in Fig.1.A flat plate is initially parallel to the streamline of the upstream flow. In order to have data to validate numerical results before step motion, the initial flow on each surface is chosen to be given by the T3A flat plate test case studied by Roach and Brierley.1For this test case, the flow is mixed laminar-turbulent with transition occurring at the point with Reynolds numberRex≈1.5×105(Fig.1).

The plate is then allowed to sink with a step motion,so that we immediately have an equivalent Angle of Attack (AoA)seen from the body-fixed frame. Both small and large amplitude step motions, with the equivalent AoAs equal to 0.25°and 5°, are considered. This situation defines the turbulent Wagner problem with transition studied in this paper.The purpose of the paper is to study how the mixed boundary layer is affected by the step motion(see Section 2)and how the force due to step motion is in turn affected by the perturbed boundary layer (see Section 3).

Fig.1 Turbulent Wagner problem with transition.

To study the present problem, the incompressible Reynolds-averaged Navier-Stokes equations are solved using the second order time accurate SIMPLE pressure-velocity coupling scheme2and the transition SST model,3for which the γ-Reθtransition model is used. The turbulent intensity and turbulent viscosity ratio are assigned 3.0%and 12 respectively at a region between 0.04 m and 0.05 m upstream of the leading edge of the plate, so that we can reproduce the experimental data of Roach and Brierley1before step motion. There are 397 grid points on each side of the plate andy+＜1.0 for the first layer adjacent to the wall. The total grid number for the whole computation domain is about 1.7×105. We first compute a steady turbulent flow without angle of attack,using a steady approach.Then we superimpose this steady flow with an upward velocity equal tovair=0.0236 m/s or 0.4724 m/s everywhere to obtain an equivalent angle of attack equal to 0.25° and 5° and start unsteady computation to simulate the step motion, with a time step, in terms of the nondimensional time τ=U∞t/cA(number of chords travelled),equal to Δτ=10-3.

2. Statement of results

First we state the results for small step motion(α=0.25°).The skin friction coefficientCfdistributions along the lower and upper surfaces at typical instants are displayed in Fig.2(a)and (b).

At τ=0, before the plate sinks,the computed skin friction matches very well with the experimental result of Roach and Brierley1for zero angle of attack. Now for τ ＞0, we observe that for the upper surface,

(1) Before τ=0.06, the transition region moves upstream.

(2) Between τ=0.06 and τ=0.17, the transition region strangely moves downstream.

(3) For τ ＞0.17, the transition region moves upstream again until it finally sets at a region between x=0.24 and x=0.55.

For the lower surface, we observe that,

(1) Between τ=0 and τ=0.05,the transition region moves upstream.

(2) Between τ=0.05 and τ=0.17, the transition region strangely moves downstream.

(3) For τ ＞0.17 the whole transition region moves downstream again until it finally sets at a region between x=0.48 and x=0.87.

Fig.2 Skin friction coefficient evolution due to a step motion for α=0.25°.

Now we reports the results for large amplitude step motion(α=5°). The skin friction distributions along the lower and upper surfaces at typical instants are displayed in Fig.3(a)and (b). For the upper surface, we observe that for τ ＞0,

(1) The laminar region globally transits to turbulent flow and at τ=0.07 the boundary layer becomes fully turbulent.

(2) For τ ＞0.70, the skin friction over some part of the leading edge becomes negative, meaning a separation bubble appears. This separation bubble is visible from the streamlines not displayed here.

Fig.3 Skin friction evolution due to a step motion for α=5°.

(3) For τ=3,the skin friction is globally lower than the initial value, due to expansion of flow separation region.

For lower surface, we observe that for τ ＞0,

(1) The transition region moves downstream immediately after the step motion before τ=0.01.

(2) Between τ=0.01 and τ=0.08, the transition region strangely moves upstream.

(3) After τ=0.08, the transition region moves downstream again.

(4) At τ=0.34,the whole boundary layer becomes laminar.

(5) At τ=0.44, the transition region appears again and starts to oscillate ever after.

3. Significance of results

The present study treated a new and interesting problem since it combines two problems of fundamental interests, one is Wagner problem and the other is boundary layer transition.

The Wagner problem for inviscid flow has important application in aeroelasticity.4Wagner problem for small AoA has been studied long ago by Wagner.5For this Wagner problem,a vortex sheet is developed downstream of the trailing edge and the force, which is initially half of the steady state value,increases gradually in time following the Wagner curve. The force reaches about 90% percent of the steady state value at non-dimensional time τ=7. These theoretical results, though based on the inviscid flow assumption, were verified experimentally for viscous flow by Walker6using the RAF130 airfoil at a Reynolds number of 1.4×104.

For large α, Graham7showed that a vortex spiral initially develops at the trailing edge, leading to a large initial peak to the force, and this force is then released and finally follows the Wagner lift curve.Li and Wu8extended the Wagner problem to moderately large angle of attack and proved that the Wagner lift curve is elevated due to the additional leading edge vortex at large angle of attack.

However, Wagner problem in the presence of an initial mixed laminar-turbulent boundary layer has not been studied before according to the knowledge of the present authors and the present study may be the first attempt to study turbulent Wagner problem with transition. In Fig.4 we display the normal force coefficientsCnas a function of the non-dimensional time for α=5°. We observe that, comparing to inviscid solution as already considered by Li and Wu,8the existence of a turbulent boundary layer reduces the magnitude of force,thought this force is still larger than predicted by the Wagner curved based on linear theory.For α=0.25°,both inviscid and turbulent solutions match well with the Wagner curve,according to our numerical simulation not displayed here. In any cases, the oscillation of the transition region appears to have not caused observable oscillation on the force.

The influence of discontinuous or step motion of the airfoil on transition has not been studied before according to the knowledge of the present authors.A related but different problem is the influence of continuous movement of airfoil on boundary layer transition, as investigated by Lee and Gerontakos9with an oscillating NACA0012 airfoil,and Richter et al.10with an oscillating EDI-M109 airfoil.They observed that under the condition when the boundary layer is mainly attached to the airfoil during the oscillation, the transition location of the upper surface moves upstream with and increasing AoA and downstream with a decreasing AoA.

Fig.4 Evolution of normal force coefficient for step motion at α=5°.

The transition behavior of boundary layer due to step motion,which is a discontinuous movement at the initial time,was shown here to display more interesting phenomena than a continuously oscillating airfoil: (A) reverse of direction of movement of transition region occurs for step motion,(B)global transition to turbulence (upper surface) and global laminarization (lower surface) are observed here. An initial discontinuous pressure gradient perpendicular to the airfoil is introduced by step motion and the mechanism by which this vertical pressure gradient affects transition remains an open question. We also wonder whether the existing transition model can correctly account for the influence of normal pressure gradient.

Large amplitude step motion can be used to understand sinking motion with fast enough acceleration,11or to model the flow of a wing interacting with a vertical gust coming from the spanwise direction. Hence, the present turbulent Wagner problem with transition cannot only be used as a pure academic problem for studying unusual unsteady flow and new phenomenon of transition, but also has real applications.

Acknowledgements

We thank the reviewer for providing us opportunity to clarify how step motion is started during the first step of computation.This work was supported by the Special Foundation of Chinese Postdoctoral Science (No. 2019T120082), Chinese Post-doc Science Foundation (No. 2018M640119), and the Natural National Science Foundation of China(No.11802157).