NUMERICAL METHOD FOR SOLVING AN INVERSE PROBLEM IN SUBSONIC FLOWS

20150186333

13/985047

September 18, 2012

It is a numerical method for solving an inverse problem about the shape design of aerodynamic body in inviscid subsonic flows. This method transfers the original Euler equations into the stream-function plane, where it solves the Riemann problem across the streamline presenting the solid-wall and obtains the geometry of the solid-wall concurrently.

1. A computer implemented numerical method for solving an inverse problem in subsonic flows, comprises the following steps: (1) transforming the two-dimensional Euler equations in the Eulerian plane, using a transforming matrix with Jacobian [mathematical expression included] into a stream-function formulation in a stream-function plane expressed by a time τ-direction, a stream-function ξ-direction and a particle traveling distance λ-direction, so-called two-dimensional Euler equations in the stream-function formulation in the stream-function plane formally are [mathematical expression included] where fS is conservation variables vector; FS and GS are respectively convection flux along the λ-direction and ξ-direction in the stream-function plane, and, [mathematical expression included] where ρ, p and E are respectively density, pressure and total energy; u, v are two velocity components in the Cartesian coordinator system; U, V are two stream-function geometry state variables; (2) building a computing grid; (3) solving so-called two-dimensional Euler equations in the stream-function formulation in the stream-function plane, including solving a Riemann problem across a streamline presenting a solid-wall.

2. The method of claim 1, wherein said computing grid is a rectangular grid constructed with the λ-direction and ξ-direction in the stream-function plane.

3. The method of claim 1, wherein said solving the time-dependent two-dimensional Euler equations in the stream-function formulation in the stream-function plane needs to literately update the conservation variable fS along the τ-direction until obtaining a steady fS

4. The method of claim 1, wherein said Riemann problem across a streamline presenting a solid-wall has the following property: there existing a state and its solid-wall-mirrored state divided by shocks or expansion waves as a left state and right state, between the two states there existing a middle state, which is divided as a left middle state and a right middle state by a contact wave presenting the solid-wall.

5. The method of claim 4, wherein said solving a Riemann problem across a streamline presenting a solid-wall has the following steps: (1) initializing flow parameters and solid-wall angle; (2) input specified pressure distribution along solid-wall; (3) finding mirror state in the Riemann problem across a streamline presenting a solid-wall; (4) calculating solid-wall angle; (5) checking convergence of the calculated solid-wall angle, and finishing if it is converged; (6) updating flow parameters; (7) repeating step (3).

6. The method of claim 5, wherein said finding mirror state in the Riemann problem across a streamline presenting a solid-wall includes the following steps: (1) Connecting the left and right states to the middle state by integrating along characteristic equations of the Euler equations in stream-function formulation; (2) Recovering velocity magnitude in the middle state; Solving a combination function f(u, v) to find flow angle in the middle state; (4) Finding the velocity component in the star state.

7. The method of claim 6, wherein said recovering velocity magnitude in the middle state is implemented according to the Rankine-Hugoniot relations across shocks and the Enthalpy constants across expansion waves.

8. The method of claim 6, wherein said combination function f(u, v) is expressed as [mathematical expression included] If pressure distribution on solid-wall is specified, it is expressed as [mathematical expression included] where pR, ρR, aR, VR, θR are known flow parameters on computing cells on solid-wall; pw is specified pressure distribution on solid-wall; θL is flow angle in solid-wall mirrored state.

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