Numerical method for simulating subsonic flows based on euler equations in lagrangian formulation
OwnerMing Luinventor
Life of this patent
Estimating the term…
Abstract
Numerical methods for simulating subsonic flows and solving inverse problems based on the new two-dimensional Euler equations in Lagrangian formulation. A transformation to derive the Euler equations in Lagrangian plane to simplify the computational grid and to minimize the numerical diffusion caused by the convective flux. Constructed is a numerical scheme, the dimensional-splitting with hybrid upwind flux operators, where the two-dimensional Riemann problem was solved in the Lagrangian plane. The method, solving the inverse geometry shape design problem in Lagrangian plane, gives the flow filed results and the geometry shape concurrently.
Claims7
A computer implemented method for simulating two-dimensional inviscid subsonic flows, comprising the following steps:
- (1) Transforming, by using a computer, a set of two-dimensional unsteady Euler equations in a Euler plane into a set of unsteady Euler equations in Lagrangian formulation in a Lagrangian plane, which has one time coordinate direction stated by a Lagrangian time τ and two space coordinate directions stated by a stream function ξ and a Lagrangian-distance λ, through a transforming matrix
- J
- =
- [
- 1
- 0
- 0
- hu
- cos
-
-
- θ
- U
- hv
- sin
-
-
- θ
- V
- ]
- ;
- said unsteady Euler equations in Lagrangian formulation include two set equations:
- a set of Lagrangian physical conservation equations and a set of geometric conservation equations, which formally are written as:
- the Lagrangian physical conservation equations
- ∂
- f
- L
- ∂
- τ
- +
- ∂
- F
- L
- ∂
- λ
- +
- ∂
- G
- L
- ∂
- ξ
- =
- 0
- ,
- where fL is the conservation variables vector and FL, GL are the convective fluxes respectively in the λ and ξ direction in the Lagrangian plane, in detailed
- f
- L
- =
- [
- ρ
-
-
- J
- ρ
-
-
- Ju
- ρ
-
-
- Jv
- ρ
-
-
- JE
- ]
- ,
- F
- L
- =
- [
- (
- 1
- -
- h
- )
-
- ρ
-
-
- K
- (
- 1
- -
- h
- )
-
- ρ
-
-
- Ku
- +
- Vp
- (
- 1
- -
- h
- )
-
- ρ
-
-
- Kv
- -
- Up
- (
- 1
- -
- h
- )
-
- ρ
-
-
- KH
- ]
- ,
- G
- L
- =
- [
- 0
- -
- p
-
-
- sin
-
-
- θ
- p
-
-
- cos
-
-
- θ
- 0
- ]
- ,
- θ
- =
- tg
- -
- 1
-
- (
- v
- u
- )
- ,
- K=uV −vU, 0<h≦1;
- the geometric conservation equations
- ∂
- f
- g
- ∂
- τ
- +
- ∂
- F
- g
- ∂
- λ
- +
- ∂
- G
- g
- ∂
- ξ
- =
- 0
- ,
-
- where
-
-
- f
- g
- =
- [
- U
- V
- ]
- ,
- F
- g
- =
- [
- 0
- 0
- ]
- ,
- G
- g
- =
- [
- -
- hu
- -
- hv
- ]
- .
- in all above, the variables t, ρ, p, E and H are respectively the time, fluid density, pressure, total energy and enthalpy;
- u, v are the Cartesian components of flow velocity V, and U, V are the Lagrangian geometry state variables;
- (2) reading object geometry for providing points on a surface of an object;
- (3) establishing a computing mesh around said object;
- (4) using a Strang-dimensional-splitting scheme with hybrid upwind flux operators solving the unsteady Euler equations in Lagrangian formulation in the Lagrangian plane.
Dependencies are inferred from the claim text.