AE 3005 HANDOUT 1

 

2-D UNSTEADY NAVIER-STOKES EQUATIONS

 

The 2-D unsteady Navier-Stokes equations may be written in a number of forms. One common form of these equations is as follows:

 

Continuity:

 

u- Momentum:

 

v-Momentum:

 

Energy Equation:

 

Here.

 

r= density; u,v = Cartesian Compnents of velocity along x,y axes;

p = Pressure ; T = Temperature.

 

Also,

 

e = Specific Internal Energy = Internal energy per unit mass of the fluid = CvT , where Cv is the specific heat at constant volume.

 

h0 = Specific Total Enthalpy = Total enthalpy per unit mass of the fluid = CpT+(u2+v2)/2 , where Cp is the specific heat at constant pressure.

 

Finally, the viscous stresses are related to the velocity field by Stokes relations

 

The molecular viscosity m and conductivity k are properties of the fluid and are functions of temperature. These two quantities are related by the Prandtl number

For air, the Prandtl number is around 0.72 at room temperatures.

 

Simplified Form for Steady, 2-D Incompressible Flows:

In steady, incompressible flows, we can drop the time derivatives because the flow is steady. The density may be also assumed constant. Then, the first three equations in the full Navier-Stokes equations set become:

 

where n is called the kinematic viscosity=m/r