Question

1. Describe the Reynolds averaging process.

2. Describe the differences between isotropic and anisotropic turbulence.

3. Describe the meaning and significance of the turbulent flow closure problem.

4.Discuss the main differences between pipe flow and flow over a flat plate in terms of Nusselt No.

5. Describe why the law-of-the wall has a direct impact on heat transfer coefficient calculations.

Answer 1

1.)

An averaging procedure applied to variable quantities such as wind speed and temperature in a turbulent flow.

If the variable is

, the averaging procedure may be given by

and

is the fluctuating part so that

where S is the average value, s the fluctuating part, and the bar indicates the averaging process. The average is usually taken over a period of time but it may be taken over space or over an ensemble of realizations. This decomposition is called Reynolds averaging. Reynolds averaging has been applied to the Navier–Stokes equations to formulate, for example, turbulent fluxes and the turbulence kinetic energy equation. The drawback of this procedure is that it leads to the problem of turbulence closure.

2.)

• Isotropic Turbulence: The statistical features have no directional preference and perfect disorder persists
• The velocity fluctuations are independent of the axis of reference, i.e. invariant to axis rotation and reflection. Isotropic turbulence is by its definition always homogeneous . In such a situation, the gradient of the mean velocity does not exist, the mean velocity is either zero or constant throughout.

In isotropic turbulence fluctuations are independent of the direction of reference and

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It is re-emphasised that even if the rms fluctuations at any point are same, their instantaneous values necessarily differ from each other at any instant.

Anisotropic Turbulence: The statistical features have directional preference and the mean velocity has a gradient.

3.)

The Navier–Stokes equations govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part. Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS) equations, which govern the mean flow. However, the nonlinearity of the Navier–Stokes equations means that the velocity fluctuations still appear in the RANS equations, in the nonlinear term from the convective acceleration. This term is known as the Reynolds stress.Its effect on the mean flow is like that of a stress term, such as from pressure or viscosity.

To obtain equations containing only the mean velocity and pressure, we need to close the RANS equations by modelling the Reynolds stress term as a function of the mean flow, removing any reference to the fluctuating part of the velocity. This is the closure problem.

4.)The flow of an unbounded fluid over a surface is treated as ‘external flow' and if the fluid is completely bounded by the surface, then it is called as ‘internal flow'. For example, flow over a flat plate is considered as external flow and flow through a pipe/duct is internal flow. However, in special cases, if the duct is partially filled and there is free surface, then it is called as open channel flow. Internal flows are dominated by viscosity whereas the viscous effects are limited to boundary layers in the solid surface for external flows.

5.)

Free convection, or natural convection, is a spontaneous flow arising from nonhomogeneous fields of volumetric (mass) forces (gravitational, centrifugal, Coriolis, electromagnetic, etc.):

If density variation Δρ is caused by spatial nonuniformity of a temperature field, then a flow arising in the Earth gravitational field is called thermal gravitation convection. The density variability may also result from nonuniform distribution of concentration of any component in a mixture or from chemical reactions, difference in phase densities or from surface tension forces at the phase interface (in this case concentration diffusion or convection is implied).