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第2题
例题详解(上）
Examples
2&3. Function relation of u(y); Navier Stokes equations; incompressible, inviscid, irrotational; vortex stretch; 2D flow
Answer:
u(y) = uw/h * y
First we take the flow as inviscid flow.
Cauchy momentum equation taking into account only action of surface forces as divergence*tau and body forces/volume, whose sum equals to rou Dv/Dt(the material derivative) ——mass/volume times acceleration following a material element, how it changes with time in space and how it changes when it moves in space. It’s the left side of the equation bellow:
On the right side, as the inviscid fluid, we only have isotropic inward-pointing pressure, which we called p, the tau is proportional to p. and the momentum equation is now
We want to consider the viscous stress using sigma,
*If a flow is incompressible this stress can be called viscous stress or deviatoric stress interchangeably. If it’s not, a compressible flow may have contribution to the deviatoric stress.
*Sigma is symmetric, as an extra stress; Newton fluid, so there is a linear relationship between sigma and f (gradient of velocity)
10 unknowns now: 3 v components, 1 pressure, 6 stress tensor sigma=sigma transpose.
Use the constitutive equation to substitute, it’s a equation of state for the viscous stress.
The Navier-Stokes equations are a statement of conservation of linear momentum for Newtonian fluid(with linear relationship between stress and velocity rate), we have constant density rou, constant isotropic viscosity miu(which is related to deformations of a constant volume.)
In general there is another term which is from the dynamic squeezing or compression and changing the volume of the material, that’s the second viscosity.
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