Question

Describe:

The characteristics of boundary layers in turbulent flows and describe which type of averaging process would most represent this flow simulation (mass versus time average, and corresponding model examples such as Crank-Nicolson, D)uFort-Frankel...)

The differences between mass and time average (give examples of each)

The attributes properties of the thin-shear layer (TSL) approximation and the attributes of boundary layer (BL) approximation.

The different driving source terms for the momentum and energy equations

Answer 1

(A) A boundary layer may be laminar or turbulent. A laminar boundary layer is one where the flow takes place in layers, i.e., each layer slides past the adjacent layers. This is in contrast to Turbulent Boundary Layers shown in Figure below.

Turbulent flow is characterized by velocity fluctuations and highly disordered motion. Most flows encountered in practice are turbulent. Turbulent flow occurs when streamlines of the liquid are irregular and change over time. The paths of the fluid flow are also irregular and form tiny whirlpool regions. The flow is turbulent when Reynolds number is greater than 4000. In practice, most flows in engineering are turbulent. However, the theory of turbulent flow remains underdeveloped since this flow is a very complex mechanism dominated by fluctuations. Therefore, turbulent flow is analysed by applying experimental measures.

For averaging process would most represent this flow simulation is corresponding model examples such as Crank-Nicolson, D)uFort-Frankel. In this mode we obtain experiment value he average velocity in experimental analysis was found to be 0.531 m/s while the average velocity from Flow Simulation depending on the boundary conditions were 0.532 m/s and. 1.375 m/s respectively.

(B) The difference is that when we increased the mass the time average is observed this is because but theoritically there isn't any relation between time and mass in speical relativity: mass is a Lorentz invariant, the same in all reference frames, time is the fourth component of the position 4-vector. Similarly, energy isn't a Lorentz invariant-it's the fourth component of the energy-momentum 4-vector. So the answer to the question is that the observed mass, which is Lorentz invariant, doesn't increase-or decrease-under Lorentz transformations. The correct formulation is that energy and momentum transform under Lorentz transformations as a 4-vector, so that E^2-(p c)^2 = (mc^2)^2 = (E')^2-(p'c)^2, where (E, p) and (E', p') are the 4-vectors in different frames, that are related through a Lorentz transformation. The expressions for the energy and the momentum, that contain the relative velocity between the two frames, express just that and nothing more: they relate the values of the energy and momentum in one frame to the values in a frame, moving with velocity beta=v/c with respect to the other, which gives rise to the factor 1/sqrt(1-beta^2)=gamma.