Researchers at Texas A&M University are studying boundary layer stability to gain insight into how transitions from laminar to turbulent flow occur. Their research has applications from low-speed flows in which the laminar-turbulent transition location affects drag and flow separation characteristics to hypersonic flows in which the difference between low laminar and much higher turbulent heating rates can determine whether a space vehicle will successfully pass through the atmosphere or burn up on reentry.

The boundary layer stability code they have developed, JoKHeR [Ref. 1], analyzes the propagation and growth of boundary layer instabilities in a pre-existing computed basic flow state. For stability analysis, it is important to create high-quality grids to compute the basic state, since boundary-layer stability is extremely sensitive to minute details and perturbations of the basic-state solution. Pointwise provides them with various ways of controlling the quality of the meshes, such as using the spacing and growth-rate functions and the elliptic PDE solver, which allows them to smooth domains and blocks.

In this study of stability and transition for subsonic aircraft, the effects of surface roughness and of roughness used for laminar flow control were examined. Discrete roughness elements (DRE) are applied as a passive control device to delay the onset of transition on swept wings [Ref. 2]. An overset grid was constructed to model a wind-tunnel experiment featuring a laminar-flow airfoil with micron-sized roughness elements applied very near the leading edge.

The height of the DRE is 24 μm, the chord of the airfoil is 1.8 m, and the wing sweep angle is 45°. The airfoil used is the ASU(67)-0315 and the experimental conditions are provided by Hunt and Saric (2011) [Ref. 3] from the Klebanoff-Saric Wind Tunnel at Texas A&M University. The airfoil model and the computational domain are shown in Figure 1.

The grid geometry consists of three structured, overset blocks: an O-grid around the circular DRE, a Cartesian path grid on top of the DRE, and a boundary layer grid. This construct is similar to that used by Rizzetta and Visbal (2007) [Ref. 4] and Rizzetta et al (2010) [Ref. 5], except the airfoil shape is no longer just an analytic shape but rather the actual coordinates of a wind-tunnel model. The DRE O-grid block size is 505x63x76 points. The DRE Cartesian patch grid is 201x201x63, and the boundary layer grid is 681x106x204. The O-grid and patch grid are shown in Figure 2, and the boundary layer grid is shown in Figure 3.

The overset grid was pre-processed with PEGASUS to compute the domain connectivity database. A high-order stencil code was used to model the flow and preliminary results have been obtained.

This work is ongoing, but preliminary results are promising. Figure 4
shows the boundary layer flow solution with DRE O-grid and patch grids
outlined. The flow is left to right and swept 45° from the chordline.
Grid slices are presented with velocity magnitude contours where deep
blue is zero and orange approaches U_{∞}.
A slice parallel to the airfoil
surface is presented at the height of the 24 μm DRE and streaks of
higher-velocity fluid can be observed from the corner and downstream
edge of the DRE. Vertical slices are also presented which show the
boundary-layer-velocity profiles as they develop with increasing chord
position. Efforts are under way to produce fully developed DRE flow
structures for input to JoKHeR. The result will be a physics-based
parabolized stability equation (PSE) model of the boundary-layer mode
shapes to be validated with the experimental measurements of Hunt and
Saric (2011) [Ref. 3].

The boundary layers over hypersonic vehicles are typically three-dimensional (3-D) and feature a variety of instability mechanisms, including viscous, acoustic, and crossflow instabilities. The Texas A&M research group studies a variety of representative geometries including straight cones, flared cones, and blunt capsules similar to the Apollo reentry capsule.

Results for the straight cone at angle of attack will be presented here. The straight cone was chosen as a verification case for JoKHeR as there is a vast amount of experimental and computational literature, including a direct numerical simulation (DNS), full Navier-Stokes computation of the instabilities by Balakumar & Owens (2010) [Ref. 6]. The cone model studied has a 7° half-angle, is 0.508 m (20 inches) in length, and has a nose radius of 0.05 mm.

Since there is no yaw in this case, only half of the cone needs to be modeled. To include the nose, a structured rectangular domain is projected onto the nose of the cone to create a domain on the surface. Figure 5 shows the mesh used in the current studies. There are two main sections in the wall-normal direction: a shock layer and a shock capture band. The shock-layer region is represented by the green color and the shock capture band is represented by the cyan color. The shock-capture band consists of 201 equally spaced points, while the shock layer contains 231 points. In the shock layer, cells are clustered toward the surface of the cone to capture the boundary layer and to the shock-capture band to ensure a smooth transition between domains. The mesh contains 274 points axially and 205 points circumferentially, totaling a mesh size of approximately 23 million cells.

Once the mesh was created with Pointwise, it was imported into GASP, a
general Navier-Stokes CFD code from
Aerosoft, Inc.
The basic state was computed at the following
conditions: freestream Mach number M_{∞} = 6,
6° angle of attack (AoA),
freestream temperature T_{∞} = 5353.42 K,
and freestream pressure P_{∞} = 11611.38 Pa abs,
which results in a unit Reynolds number Re′ =
1010.09×10^{6} per meter.
An isothermal wall-temperature condition, T_{wall} =
300 K, was imposed. Figure 6 shows the basic-state Mach contour solution
obtained from GASP.

In using PSE, a marching path must be specified in order to compute boundary-layer stability along that path. While inviscid streamlines provide a convenient marching path, they can be different from the true direction of disturbance propagation as defined by the group velocity. However, computing the group velocity requires a whole new set of stability calculations. On the straight cone at yaw, off the windward and leeward rays, the dominant instability is crossflow. Stationary crossflow vortices do not follow inviscid streamlines, but if one is familiar with the physics of the problem, one can implement an approximate method to find the stationary crossflow vortices based solely on the undisturbed basic-state solution. Figure 7 shows JoKHeR computed N-factors (natural log of integrated disturbance growth) along different computed crossflow stationary vortex paths compared to the DNS results. The red symbols represent N-factors computed from the DNS of Balakumar & Owens (2010) [Ref. 6]. The black lines represent the N-factors computed by using an eight million cell mesh without nose tip resolution for three different stationary crossflow vortices that correspond to the vortices in the DNS. The blue line represents the N-factor results obtained with the current 23 million cell mesh. Good agreement was obtained between JoKHeR results and the DNS in the verification studies, as is evident by comparing the black lines and red symbols. The agreement improved when N-factors were recomputed with the 23 million cell mesh basic state. This is significant because JoKHeR uses a fraction of the computational resources required for the DNS.

Boundary layer stability prediction aids understanding of boundary layer transition in a variety of flow situations from very low speeds up to hypersonic speeds. The parabolized stability equation code, JoKHeR, developed at Texas A&M University provides an efficient way to make these predictions, but it does require an accurate basic flowfield state as a basis, and this in turn requires a good computational grid. Pointwise gives researchers the control they need to meet their demanding basic-state accuracy requirements and so predict how boundary layer disturbances will propagate and possibly cause a laminar flow to transition to turbulent.

- Kuehl, J., Perez, E., and Reed, H. 2012. JoKHeR: NPSE Simulations of Hypersonic Crossflow Instability. AIAA-2012-0921.
- Saric, W., Carpenter, A., and Reed, H. 2011. Passive Control of Transition with Roughness in Three-Dimensional Boundary Layers. Philosophical Transactions of the Royal Society A, vol. 369, no. 1940, 1352-1364, April 2011.
- Hunt L. E., and Saric W. S. 2011. Boundary-Layer Receptivity of Three-Dimensional Roughness Arrays on a Swept Wing. AIAA-2011-3881.
- Rizzetta, D. P., and Visbal, M. R., "Direct Numerical Simulation of Flow Past an Array of Distributed Roughness Elements," AIAA Journal vol. 45, no. 8, August 2007, pp. 1967-1976. doi: 10.2514/1.25916
- Rizzetta, D. P., Visbal, M. R., Reed, H. L., and Saric, W. S., "Direct Numerical Simulation of Discrete Roughness on a Swept-Wing Leading Edge," AIAA Journal vol. 48, no. 11, November 2010, pp. 2660-2673. doi: 10.2514/1.J050548
- Balakumar P, Owens LR. 2010. Stability of hypersonic boundary layers on a cone at angle of attack. AIAA-2010-4718.

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