Dissipation versus pressure drop
 

Hi, I took the DNS data by Mansour,Kim,Moin (1988) of a turbulent channel flow Re=3250, Retau=178.12. I thought the integral of the Dissipation (viscious+turbulent) must equal the pressure drop times vol. flux (energy balance) Pressure Drop: dp/dx=tw/H (1) (tw=wall shear stress, H=half channel width) Energy balance: dp/dx * Um * H = int_0^(H) DIS dy + int_0^(H) rho * epsilon *dy (2) (Um=mean Velocity, DIS=viscious Dissipation, epsilon=turbulent Dissipation). In dimensionless Form the energy balance can be written as: Re/Retau**2 = int_0^1 DIS* dy* + int_0^1 epsilon* dy* (3) (dimensionless by nu/utau**4). By integrating the DNS data, I obtain: 0.08676 for the dissipation and 0.102437 for the pressure drop times vol. flux. Why is there a 15% difference? I checked formular (3) for laminar flow and the balance was right. Thanks Fabian

Re: Dissipation versus pressure drop
 

What formula did you use for turbulent dissipation.

Turbulent dissipation is 2*nu*{s_ij*s_ij}, where {.} indicates an ensemble average. Sometimes, the term nu*{ (du_i/dx_j)* (du_i/dx_j) } is also referred to as dissipation but isn't quite right. There needs to be another term nu*{ (du_i/dx_j)* (du_j/dx_i) } which supposedly has more dispersive behavior than dissipative and is hence dropped out. However, the sum of the 2 terms is the real dissipation. Only when you include both these terms, you have a definition for dissipation which is material frame indifferent. If the 2nd term is not included, you have invariance under linear translational motions but not under arbitrary rigid body motions (like rigid body rotation).

If you have already use the right formula and still have a mismatch, I suggest you use the same spatial stencils (discretization) as those used in the simulations. This has to do with discrete conservation/compatibility. Also one should note that, although the mean and rms profiles for velocities in a DNS might match the experimental data, accurate prediction of wall stresses is not guaranteed unless the viscous sub-layer is well resolved. Typically, min (dy+} < 0.2-0.3.