计算流体力学.pdf

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Introduction to typical types inertia or acceleration
component in the momentum equation. These terms are called advection terms. The
energy equation has nearly similar terms, usually called the convection terms, which
involve the motion of the flow field. For unsteady two-dimensional problems, the
appropriate equation can be represented as
∂φ ∂φ ∂φ "∂2φ ∂2φ %
+ u + v = B$ + '+ S
∂t ∂x ∂y #∂x2 ∂y2 &
φ denotes velocity, temperature or some other transported property,
u and v are velocity components,
B is the diffusivity for momentum or heat, and
S is a source term.
The pressure gradients in the momentum or the volumetric heating in the energy
equation can be appropriately substituted in S.
The above PDE is parabolic in time and elliptic in space.
For very high-speed flows, the terms on the left side dominate, the second-order terms on
the right hand side become trivial, and the equation become hyperbolic in time and 2D Euler equation
∂u ∂u ∂u 1 ∂p
+ u + v = − + f
∂t ∂x ∂y ρ ∂x x
∂v ∂v ∂v 1 ∂p
+ u + v = − + f
∂t ∂x ∂y ρ ∂y y
∂φ ∂φ
u = , v = −
∂y ∂x
2 2 2 2
∂ #∂u ∂v & ∂u ∂u ∂ u ∂u ∂v ∂ u ∂u ∂v ∂ v ∂v ∂v ∂ v ∂f ∂fy
% − (+ + u + + v − − u − − v = x −
∂t $∂y ∂x ' ∂y ∂x ∂x∂y ∂y ∂y ∂y2 ∂x ∂x ∂x2 ∂x ∂y ∂x∂y ∂y ∂x
2 2
∂ u ∂ u ∂f ∂fy 2 2
→ v + u +....... = x − B − 4AC = u > 0, the equation is Hyperbolic in space
∂y2 ∂x∂y ∂y ∂xBoundary conditions
The spatial boundary conditions in flow and heat transfer problems are of three general
types. They may stated as
on boundary: φ = φ1 (x) → Dirichlet BC
∂φ
on boundary: = φ2 (x) → Neumann B