The incompressible Navier-Stokes equation for a fluid of constant density and…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Instability, transition and turbulence · Instability, transition and turbulence

Fluid Dynamics, Ch Transition to turbulence

Problem

The incompressible Navier-Stokes equation for a fluid of constant density ρ\rho and kinematic viscosity ν\nu can be written in momentum-flux form as ∂ui∂t+∂∂xj(uiuj)=−1ρ∂p∂xi+ν∂2ui∂xj∂xj,∂ui∂xi=0.\frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j}(u_iu_j) = -\frac{1}{\rho}\frac{\partial p}{\partial x_i} + \nu\frac{\partial^2 u_i}{\partial x_j\partial x_j}, \qquad \frac{\partial u_i}{\partial x_i} = 0. Write each field as a Reynolds decomposition about a time average denoted by an overbar, ui=uˉi+ui′u_i = \bar u_i + u_i' and p=pˉ+p′p=\bar p+p', with ui′‾=0\overline{u_i'}=0, p′‾=0\overline{p'}=0 and aˉ‾=aˉ\overline{\bar a}=\bar a for any field aa. ((a)) Average the continuity equation and show that the mean velocity field is itself divergence-free. ((b)) Average the momentum equation, using uiuj‾=uˉiuˉj+ui′uj′‾\overline{u_iu_j} = \bar u_i\bar u_j + \overline{u_i'u_j'}, to obtain the Reynolds-averaged momentum equation. Identify the term that was not present in the original equation. ((c)) For a general unsteady, three-dimensional turbulent flow, count the independent unknown mean fields (the components of uˉi\bar u_i, pˉ\bar p, and the independent components of the symmetric tensor ui′uj′‾\overline{u_i'u_j'}) and the independent equations available from parts (a) and (b) (mean continuity and the three components of mean momentum). State by how many the unknowns exceed the equations.

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