[srinivasvl81]

In TKE equation, the dissipation term can be written as:

= -2nu*<s_ij s_ij>

where, nu=mu/rho is kinematic viscosity, s_ij= 0.5*{ du_i/dx_j + du_j/dx_i }, mu=dynamic viscosity, rho=density.

here, u= fluctuating velocity component, d=partial derivative operator, <>=time average

Considering a problem of sub-meso scale ocean modelling,

1. Is the density, instantaneous or fluctuating or mean ?

2. After doing a time average<> on the strain rate tensors, s_ij there will be no time dimension left, if the density is instantaneous, will the <s_ij s_ij> be constant over all time steps of density ?

3. If the density is constant, will be dissipation also be constant?

[FMDenaro]

Originally Posted by srinivasvl81

In TKE equation, the dissipation term can be written as:

= -2nu*<s_ij s_ij>

where, nu=mu/rho is kinematic viscosity, s_ij= 0.5*{ du_i/dx_j + du_j/dx_i }, mu=dynamic viscosity, rho=density.

here, u= fluctuating velocity component, d=partial derivative operator, <>=time average

Considering a problem of sub-meso scale ocean modelling,

1. Is the density, instantaneous or fluctuating or mean ?

2. After doing a time average<> on the strain rate tensors, s_ij there will be no time dimension left, if the density is instantaneous, will the <s_ij s_ij> be constant over all time steps of density ?

3. If the density is constant, will be dissipation also be constant?

The general analysis you find for the turbulence has a hystorical assumption, that is the density is homogeneous. For the ocean it can be introduced the Bousinnesq approximation for the bouyancy to take into account the dependence of density on low temperature gradients (actually you could also add the salinity gradient). However, the kinematic viscosity is assumed to be constant.

Have a look here for the ocean flow model https://www.researchgate.net/publica...physical_flows


Note that for general non-constant density flow problem, the Favre averaging is introduced.

[LuckyTran]

nu also needs to be averaged. Your formula implicitly assumes nu is a constant. Use a formula that doesn't assume this and a lot of your confusion will be cleared away.

Btw, Reynolds averaging works also for unsteady flows, you can extend the <> operation beyond time-averaging and allow for time-dependent flows. Think URANS!

However, the answer is....

1. Density is the reynolds average (which can be the mean, but can also be time-dependent like in unsteady RANS). But it won't be instantaneous, otherwise the TKE equation would not be a TKE equation.

[srinivasvl81]

Thank you for the replies.

By the way, in the <s_ij s_ij> term, is the multiplication, a dot product or a normal matrix product ?

[FMDenaro]

Originally Posted by srinivasvl81

Thank you for the replies.

By the way, in the <s_ij s_ij> term, is the multiplication, a dot product or a normal matrix product ?

It stands for <S:S>, that means you perform the doct product twice