Hi

In the horizontal momemtum equation written in this form

Dt(v) + div(Vv) + Dz(Vw) + f(k X v) + grad p = Q

where Dt(v) is partial differential of momemtum density w.r.t time t , grad p is the pressure gradient, Q is source/sink due to mass/force ,f is the Coriolis parameter and v = V/rho is the momemtum density,

what does div(Vv) stand for? Why should that term be there?

Thanks a bunch

Hi

The term div(Vv) is link to lagrangian derivation (i.e. derivation along streamlines).

From a formal point of view, the momentum equation comes from a conservation equation with an integral form (Newton principle): d_t{INT_omega(t)[rho U]dV}=INT_omega(t)[volume forces]dV+INT_sigma(t)[surface forces]dS (1) where INT means integration, omega(t) is a volume varying with time and sigma(t) the boundary of this volume.

Because omega is time depending, you cannot permute d_t and INT_omega(t) directly. The correct way to do this is to use the following formulation d_t{INT_omega(t)[A]dV}=INT_omega(t)[d_t(A)]dV+INT_sigma(t)[A*U]dS, where A is a vector and * is a tensorial product

The equation (1) then becomes : INT_omega(t)[d_t(rho U)]dV+INT_sigma(t)[rho]dS =INT_omega(t)[volume forces]dV+INT_sigma(t)[surface forces]dS

Using Green's theorem ( INT_sigma[A]dS =INT_omega[DIV(A)]dV with DIV the divergence of a tensor) we can write: INT_sigma{ d_t(rho U)+DIV((rho U)*U)}dV= INT_sigma{ Volume force + DIV(Surface forces)}dV

Now we assume that the fluid is newtonian with viscosity equal to 0, that the density rho is constant (incompressible) and that we're in a rotating frame. Then, suppressing the INT_, the equation (vector formulation) writes : d_t(U)+DIV((rho U)*U)=-grad(p)/rho + Coriolis + additionnal volume forces + div(additional surface forces)

Projecting this along an axis (X for instance), we obtain (div is the divergence of a vector): d_t(u)+div(uU)=d_t(u)+d_x(u^2)+d_y(uv)+d_z(uw)= -d_x(p)/rho + (...).X, where u,v,w are the three components of the velocity vector U.

Best regards

Dear Number

This term correspond to the transfer of momentum by convective motion inside the fluid.

This term should be there only if you want to write the real Navier-Stokes eq. or if it is comparable (in order of magnitude) to the rest of the term into the equation.

As you can see this is a non-linear term and it is maybe one of the more important reasons (personal opinion) why we are here in this forum talking about COMPUTATIONAL Fluid Dynamics.

Regards

Kike

You can check the book of Prof. Slattery: Momentum, Energy, and Mass Transfer in Continua. This is a result of transport theorem for a region containing a singular surcafe.