Shock detachment from curved wedges by local choking: numerical verification

Abstract

Computational fluid dynamics shows that a shock wave can detach from the sharp leading edge of a curved wedge at a wedge angle smaller than the classical maximum flow deflection as well as the sonic wedge angle. This is attributed to the inability of the sonic flow, at the wedge trailing edge, to pass as much mass flow as is being admitted through the shock wave attached at the leading edge. At this condition, the flow is unsteady, causing both the sonic surface and the shock to make adjustments in their shapes and positions to achieve a steady state with mass-flow balance. As a result, the shock wave becomes detached. Time-accurate CFD calculations show the gasdynamic details of the adjustment where the flow and the detached shock assume a steady state as the mass-flow imbalance gradually decreases to zero. This mechanism of shock detachment, occurring near the leading edge, is called local choking to distinguish it from shock detachment due to global choking that occurs because of flow choking at the exit of a convergent duct and to distinguish it as well from detachment due to an excessive leading-edge deflection. The local choking mechanism has been postulated to be a cause of shock detachment from doubly curved wedges. An analysis, based on curved shock theory and confirmed by CFD, shows that local choking and shock detachment from a doubly curved leading edge are dependent on Mach number, wedge angle, wedge curvature (both streamwise and cross-stream), and wedge length.

Appendices

Appendix 1: Determination of shock attachment/detachment state by Fluent

In order to verify the ability of Fluent to accurately distinguish between the attached/detached state of the shock, a test calculation was performed on two plane wedges as shown in Fig. 17. Flows on the upper and lower wedges are isolated by a splitter plate so as to prevent any interaction between the upper and lower flows. Plane wedges were chosen for testing because conditions at sonic or Crocco points present no singularities or anomalies on a flat shock so that detachment can be expected to be “hard” at the maximum deflection condition. At Mach 3, the theoretical maximum flow deflection angle is $34.0734398^{\circ }$. The deflection of the upper plate is set at $0.01^{\circ }$ less, and the deflection of the lower plate is $0.01^{\circ }$ more than this value. CFD results, in Fig. 17, show that the upper shock is attached and the lower shock is detached. The lower shock standoff distance, divided by the wedge length, is $1.328 \times 10^{-5}$. The upper and lower surface Mach number and pressure distributions are shown in Fig. 18. Significant differences are apparent in the attached and detached shock cases. Fluent correctly predicts and displays an attached shock on the upper surface and a detached shock on the lower surface of the flat wedges so that Fluent is capable of distinguishing and displaying the attached/detached state of a shock with an accuracy of at least $0.02^{\circ }$ in flow deflection angle. This accuracy is sufficient to resolve the flows at the sonic, Crocco, maximum deflection, and Thomas points, these being separated by more than $0.01^{\circ }$ of flow deflection. It is expected that this sensitivity is maintained for shock attachment/detachment detection on curved wedges as well.

Fig. 18

Mach number and pressure ratio along the splitter plate and on each wedge

Fig. 19

Domain geometry

Fig. 20

Gradient adaption used on Mach number

Appendix 2: Numerical simulation tool

Ansys Fluent 18.1 was used for the numerical simulations. The numerical simulations were conducted on 2D (axisymmetric and planar), inviscid, ideal gas (air) using the steady-state density-based solver.

1.1 Domain geometry

The domain geometry is shown in Fig. 19. The leading edge of the body under test always started at (x, $y)=(0,1000)\,\hbox {mm}$ with sufficient length of LE Wall to capture a detached shock.

Both the inlet and the outlet were set as pressure far-field boundary condition. The pressure far-field boundary condition is often called a characteristic boundary condition, since it uses characteristic information (Riemann invariants) to determine the flow variables at the boundaries. The free stream conditions used were $M = 3.0$, $P = 1\,\hbox {atm}$, $T = 288.16\,\hbox {K}$. The LE wall, body, and TE wall were set as wall boundary conditions (no inflow or outflow). The symmetry condition was imposed at the lower boundary domain.

1.2 Meshing

Initial meshing condition used $0.5\,\hbox {mm}$ length for the body, $1\,\hbox {mm}$ for the LE wall and TE wall, and $25\,\hbox {mm}$ for the inlet, outlet, and symmetry. These conditions represented the coarsest mesh imposed on the domain as the study used Fluent’s gradient adaption which performs automatic refinement or coarsening of the mesh based on gradients of Mach number. Figure 20 shows a typical solution of mesh adaptation based on gradients of Mach number.

1.3 Solution method

The method selected for all studies was: implicit, Roe-FDS (flux-difference splitting scheme). The study also used Fluent’s solution steering which is described in the Fluent manual as follows: “solution steering in the density-based implicit solver provides you with an expert system that will help navigate the flow solution from a starting initial guess to a converged solution with minimum user interaction. When you apply solution steering, you will be required to select the type of flow that best characterizes the solution domain and the maximum desired accuracy and then allow the solver to take the solution to convergence. As the solver proceeds with the solution iteration, certain solver parameters will be adjusted behind the scenes to ensure that a converged solution to steady-state is possible” [8].

Solution steering will perform full multigrid (FMG) initialization followed by two iterative stages:

Stage 1: During this stage, the solution is advanced gradually from first-order accuracy to a user specified higher-order accuracy at a constant low CFL value (CFL $=$ 0.1 used for this study). For this study, we maintained first order (i.e., used 0% of second order);

Stage 2: The solution is driven towards convergence by regular adjustments of the CFL value up to a maximum, user specified CFL value (CFL $=$ 2.0 used for this study). Fluent automatically adjusts the CFL value during the iteration based on the residual history. For increasing residuals, the CFL is decreased and vice versa.

1.4 Convergence criteria

For this study, the steady-state convergence criteria used the following:

• 1. Difference between mass flow at the inlet and mass flow at the exit was $0.001\,\hbox {kg}/\hbox {s}$ or less and;

• 2. Automatic grid adaption did not result in a change of total cells produced. There were no coarsening of grids and no grid refinement in the calculation.

• 3. All residuals (continuity, velocity, and energy) stopped decreasing and maintained a constant state after iterations. This is related to item 2 above in Fluent’s use of grid adaption.

• 4. The solution structure (specifically the shock structure) remained stationary. There was no change in the flow field calculated.