Literature review: RANS derivation and analysis
INTRODUCTION:
"Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another. "
Turbulence is a continuous phenomenon that exists on a large range of length and time scales, which are still larger than molecular scales. In order to visualize turbulent flows, one often refers to turbulent eddies, which can be thought of as a local swirling motion whose characteristic dimension is on the order of the local turbulence length scale. Turbulent eddies also overlap in space, where larger eddies carry smaller ones. As there exists a large range of different scales (or turbulent eddy sizes), an energy cascade exists by which energy is transferred from the larger scales to the smaller scales, and eventually to the smallest scales where the energy is dissipated into heat by molecular viscosity.
Turbulent flows are thus always dissipative.Turbulent flows also exhibit a largely enhanced diffusivity. This turbulent diffusion greatly enhances the transfer of mass, momentum, and energy. The apparent stresses, therefore, may be of several orders of magnitude greater than in the corresponding laminar case. The fact that the Navier-Stokes equations are non-linear for turbulent flows leads to interactions
between fluctuations of different wavelengths and directions. The wavelengths of the motion may be as large as a characteristic scale on the order of the width of the flow, all the way to the smallest scales, which are limited by the viscous dissipation of energy. The action of vortex stretching is mainly responsible for spreading the motion over a wide range of wavelengths.
Wavelengths which are nearly comparable to the characteristic mean-flow scales interact most strongly with the mean flow. This implies that the larger-scale turbulent eddies are most responsible for the energy transfer and enhanced diffusivity. In turn, these large eddies cause random stretching of the vortex elements of the smaller eddies, and energy is cascades down from the largest to the smallest scales.
Direct Numerical Simulation:
Numerical methods that do not employ any averaging and turbulent models – known as Direct Numerical Simulation (DNS) have been developed. The DNS does not depend on any turbulent model (often empirical) and are based on the local instantaneous governing equations (1) – (6). The DNS is the closest to the real transport phenomena, but it is computationally most intensive. Since DNS is not practical for many engineering problems, alternative approaches like Reynolds Averaged Numerical Solution (RANS) or Large Eddy Simulation (LES) can be employed. For academic research, on the other hand, DNS can reproduce experimental resolution for experiments that are either very difficult or impossible. With advancements on parallel computing, one can break up the computational task and set to multiple CPUs to perform computation simultaneously. The scale of the number of grid points required for a DNS to capture the physics of the simplest turbulent flow is
`N approx L/eta`
where, L is the length scale of the largest turbulent eddy
`eta` is the length scale of the viscous effects.
The length scale of viscous effect is also referred as Komogorov Scale, and is given as:
`eta = (nu^3/(epsi))^(1/4)`
where `nu` is the kinematic viscosity and `sigma` is the rate of kinetic energy dissipation. It has been shown that the scale of `L/eta` is:
`N ~ L/eta ~ (Re_L)^(3/4)`
where `Re_L` is the turbulent Reynolds number.
When DNS is employed to solvea three-dimensional turbulent problem, the number of grid required will be:
`N^3~(Re_L)^(9/4)`
which shows that the DNS can easily run into computational limitation of speed and memory for large Reynolds numbers. Nevertheless, significant progresses have been made in DNS of turbulent flow especially turbulent boundary layer structures.
Time - Averaged Governing Equations for Turbulence:
The transport phenomena equations including the Navier-Strokes and the energy equation are exact for solving the turbulent problem if their complete unsteady forms are solved. In general, this approach is not taken because of complexity and significant computer time requirement. In addition, the turbulent problems, in their gross aspect, are assumed steady state as noted in the previous subsection.
Time averaging of transport phenomena equations should provide the net effect of the turbulent perturbation. For an incompressible binary system with constant properties, the continuity, Navier-Stokes, energy, and conservation of mass species equations in a Cartesian coordinate system are
`(del u)/(del x) + (del v)/(del y) + (delw)/(del z) = 0` ----(1)
`(del u)/(del t) + u(del u)/(del x) + v(del u)/(del y)+w(del u)/(del z) = -1/(rho)(del p)/(del x) + nu ((del^2u)/(delx^2) + (del^2u)/(dely^2) + (del^2u)/(delz^2))` ----(2)
`(del v)/(del t) + u(del v)/(del x) + v(del v)/(del y)+w(del v)/(del z) = -1/(rho)(del p)/(del x) + nu ((del^2v)/(delx^2) + (del^2v)/(dely^2) + (del^2v)/(delz^2))` ----(3)
`(del w)/(del t) + u(del w)/(del x) + v(del w)/(del y)+w(del w)/(del z) = -1/(rho)(del p)/(del x) + nu ((del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2))` ----(4)
`(delT)/(delt) + u(delT)/(delx) + v(delT)/(dely)+w(delT)/(delz) = alpha((del^2T)/(delx^2) + (del^2T)/(dely^2) + (del^2T)/(delz^2))` ----(5)
`(del omega)/(delt) + u(del omega)/(delx) + v(del omega)/(dely)+w(del omega)/(delz) = alpha((del^2 omega)/(delx^2) + (del^2 omega)/(dely^2) + (del^2 omega)/(delz^2))` ----(6)
Since we made no assumption about the nature of the flow in the above equations, the local instantaneous parameters in a turbulent flow satisfy eqs. (1) – (6). However, applying the above equations to turbulent flow will require very fine grid size to be able to capture the detailed flow structure within each eddy – which is very challenging and, in most cases, unnecessary. For most practical engineers, the mean values of various parameters in a turbulent flow are of interest. Therefore, we will obtain the governing equations in term of time-averaged parameters. For turbulent flow, the velocity components, pressure, temperature and mass fraction of species can be expressed as sums of their mean values and fluctuations
`u = bar u (x,y,z,t) + u'(x,y,z,t)`
`v = bar v (x,y,z,t) + v'(x,y,z,t)`
`w = bar w (x,y,z,t) + w'(x,y,z,t)`
`p = bar p (x,y,z,t) + p'(x,y,z,t)`
`T = bar T (x,y,z,t) + T'(x,y,z,t)`
`omega = bar omega (x,y,z,t) + omega'(x,y,z,t)`
1. Continuity Equation:
Substituting the above equations into the continuity equation (1), we get:
`del/(del x) (bar u + u') + del/(del y) (bar v + v') + del/(del z) (bar w + w') = 0` ----(7)
By taking time averaging of eq. (7), we have:
`bar(del/(del x) (bar u + u')) + bar(del/(del y) (bar v + v')) + bar(del/(del z) (bar w + w')) = 0` ----(8)
Considering eq. `bar(bar phi bar psi) = bar(bar psi bar phi), bar(bar phi phi') = bar(phi phi') = 0, bar(bar phi + bar psi) = bar phi + bar psi`, the continuity equation (8) becomes:
`(del bar u)/(del x) + (del bar v)/(del y) + (del bar w)/(del z) = 0` ----(9)
which is the continuity equation in terms of time averaged velocity. It can be seen that it has the exactly same form as that in terms of local instantaneous velocity, eq. (1). In other words, the contributions of the fluctuations of velocity components are averaged out during the time averaging.
2. Momentum Equation:
Similarly, substituting
`u = bar u (x,y,z,t) + u'(x,y,z,t)`
`v = bar v (x,y,z,t) + v'(x,y,z,t)`
`w = bar w (x,y,z,t) + w'(x,y,z,t)`
`p = bar p (x,y,z,t) + p'(x,y,z,t)`
into the momentum equation (2) in the x-direction and taking time average, we get
`bar(del bar u)/(del t) + bar(del u')/(del t) + bar(bar u (del bar u)/(del x)) + bar(bar u (del u')/(del x)) + bar(u' (del bar u)/(del x)) + bar(u' (del u')/(del x)) + bar(bar v (del bar u)/(del y)) + bar(bar v (del u')/(del y)) + bar(v' (del bar u)/(del y)) + bar(v' (del u')/(del y))+bar(bar w(del bar u)/(del z)) + bar(bar w(del u')/(del z)) + bar(w' (del bar u)/(del z)) + bar(w' (del u')/(del z)) = -1/rho bar((del bar p)/(del x)) -1/rho bar((del p')/(del x)) + nu (bar(del^2 bar u)/(del x^2)+bar(del^2 u')/(del x^2)+bar(del^2 bar u)/(del y^2)+bar(del^2 u')/(del y^2)+bar(del^2 bar u)/(del z^2)+bar(del^2 u')/(del z^2))` ----(10)
Considering eq. `bar(bar phi bar psi) = bar(bar psi bar phi), bar(bar phi phi') = bar(phi phi') = 0, bar(bar phi + bar psi) = bar phi + bar psi`, the momentum equation in x-direction becomes:
`(del bar u)/(del t) + bar u (del bar u)/(del x) + bar v (del bar u)/(del y) + bar w (del bar u)/(del z) = -1/rho (del bar p)/(del x) + nu ((del^2 bar u)/(del x^2) + (del^2 bar u)/(del y^2) + (del^2 bar u)/(del z^2)) - (bar(u' (del u')/(del x)) + bar(v' (del u')/(del y)) + bar(w' (del u')/(del z)))` ----(11)
where fluctuation components only appear in the last parenthesis on the right-hand side. In order to simplify the terms that contain fluctuation component, the continuity equation (1) is multiplied by u, i.e.,
`u (del u)/(del x) + u (del v)/(del y) + u (del w)/(del z) = 0`
Substituting eqs,
`u = bar u (x,y,z,t) + u'(x,y,z,t)`
`v = bar v (x,y,z,t) + v'(x,y,z,t)`
`w = bar w (x,y,z,t) + w'(x,y,z,t)`
`p = bar p (x,y,z,t) + p'(x,y,z,t)`
from Description of turbulence into the above equation and performing time averaging to the resultant equation, we have:
`bar(bar u (del bar u)/(del x)) + bar(bar u (del u')/(del x)) + bar(u' (del bar u)/(del x)) + bar(u' (del u')/(del x))+bar(bar u (del bar v)/(del y)) + bar(bar u (del v')/(del y)) + bar(u' (del bar v)/(del y)) + bar(u' (del v')/(del y))+bar(bar u (del bar w)/(del z)) + bar(bar u (del w')/(del z)) + bar(u' (del bar w)/(del z)) + bar(u' (del w')/(del z)) = 0`
which can be simplified to
`bar u (del bar u)/(del x) + bar u (del bar v)/(del y) + bar u (del bar w)/(del z) + bar(u' (del u')/(del x) )+bar(u' (del v')/(del y) )+bar(u' (del w')/(del z) ) = 0`
Considering the continuity equation in terms of mean velocity component, eq. (9), the above equation is further simplified to:
`bar(u' (del u')/(del x) )+bar(u' (del v')/(del y) )+bar(u' (del w')/(del z) ) = 0` ----(12)
Substituting the eq. (12) to eq. (11), the momentum equation in the x-direction becomes
`(del bar u)/(del t) + bar u (del bar u)/(del x) + bar v (del bar u)/(del y) + bar w (del bar u)/(del z) = -1/rho (del bar p)/(del x) + nu ((del^2 bar u)/(del x^2)+(del^2 bar u)/(del y^2)+(del^2 bar u)/(del z^2)) - ((del bar ((u')^2))/(del x)+(del bar ((v'u')))/(del y) + (del bar ((w'u')))/(del z))` ----(13)
By following the same procedure, the momentum equation in the y- and z- directions become:
`(del bar v)/(del t) + bar u (del bar v)/(del x) + bar v (del bar v)/(del y) + bar w (del bar v)/(del z) = -1/rho (del bar p)/(del y) + nu ((del^2 bar v)/(del x^2)+(del^2 bar v)/(del y^2)+(del^2 bar v)/(del z^2)) - ((del bar ((u'v')))/(del x)+(del bar ((v')^2))/(del y) + (del bar ((w'v')))/(del z))` ----(14)
`(del bar w)/(del t) + bar u (del bar w)/(del x) + bar v (del bar w)/(del y) + bar w (del bar w)/(del z) = -1/rho (del bar p)/(del z) + nu ((del^2 bar w)/(del x^2)+(del^2 bar w)/(del y^2)+(del^2 bar w)/(del z^2)) - ((del bar ((u'w')))/(del x)+(del bar ((v'w')))/(del y) + (del bar ((w')^2))/(del z))` ----(15)
It can be seen from eqs. (13) – (15) that extra terms associated with the fluctuations of the velocity components appeared in the momentum equations. These extra terms account for the momentum transfer caused by the velocity fluctuations and are termed as turbulent or Reynolds stress terms. The difference between the viscous stress and Reynolds stress is that the former occurs at the molecular scale whereas the latter is caused by the interaction between eddies.
3. Energy Equation:
To obtain the energy equation in terms of mean values of various variables, the following equations:
`u = bar u (x,y,z,t) + u'(x,y,z,t)`
`v = bar v (x,y,z,t) + v'(x,y,z,t)`
`w = bar w (x,y,z,t) + w'(x,y,z,t)`
`p = bar p (x,y,z,t) + p'(x,y,z,t)`
`T = bar T (x,y,z,t) + T'(x,y,z,t)`
are substituted into the energy equation (5) and time averaging is performed to the resultant equations to obtain
`bar(del barT)/(del t) + bar(del T')/(del t) + bar(bar u (del bar T)/(del x)) + bar(bar u (del T')/(del x)) + bar(u' (del bar T)/(del x)) + bar(u' (del T')/(del x)) + bar(bar v (del bar T)/(del y)) + bar(bar v (del T')/(del y)) + bar(v' (del bar T)/(del y)) + bar(v' (del T')/(del y)) + bar(bar w (del bar T)/(del z)) + bar(bar w (del T')/(del z)) + bar(w' (del bar T)/(del z)) + bar(w' (del T')/(del z)) = alpha ( bar(del^2 bar T)/(del x^2) + bar(del^2 T')/(del x^2) + bar(del^2 bar T)/(del y^2) + bar(del^2 T')/(del y^2) + bar(del^2 bar T)/(del z^2) + bar(del^2 T')/(del z^2))` ----(16)
Considering eq. `bar(bar phi bar psi) = bar(bar psi bar phi), bar(bar phi phi') = bar(phi phi') = 0, bar(bar phi + bar psi) = bar phi + bar psi`, the energy equation becomes:
`(del bar T)/(del t) + bar u (del bar T)/(del x) + bar v (del bar T)/(del y) + bar w (del bar T)/(del z) = alpha ( (del^2 bar T)/(del x^2) + (del^2 bar T)/(del y^2) + (del^2 bar T)/(del z^2)) - ( bar (u' (del T')/(del x)) + bar (v' (del T')/(del y)) + bar (w' (del T')/(del z)) )` ----(17)
By following the similar procedure that eq. (12) was derived, the following equation can be obtained:
`bar(T' (del u')/(del x)) + bar(T' (del v')/(del y)) + bar(T' (del w')/(del z)) = 0` ----(18)
Substituting the above equation into eq. (17), the energy equation becomes
`(del bar T)/(del t) + bar u (del bar T)/(del x) + bar v (del bar T)/(del y) + bar w (del bar T)/(del z) = alpha( (del^2 bar T)/(del x^2) + (del^2 bar T)/(del y^2) + (del^2 bar T)/(del z^2)) - ( (del bar(u'T'))/(del x) + (del bar(v'T'))/(del y) + (del bar(w'T'))/(del z) )` ----(19)
Reynolds Stress:
Assume a flow `vec u` with a time scale T. Let `tau` denote a time scale `tau` << T. We can then write for each component of the velocity
`u_i - bar u_i + u_i'` ----(1)
where by definition
`bar u_i = 1/ tau int_o^tau u_i dt`
It immediately follows that
`bar(u_i') = bar (u_i - bar(u_i)) = bar u_i - bar u_i = 0`, also `del /(del x) bar u_i = bar( (del u_i)/(del x)` etc.
Substitute equation (1) into continuity and average over `tau`, i.e., talk `bar(( ))`
`bar((del u_i)/(del x_i)) = bar((del bar u_i)/(del x_i)) + ubrace (bar(del u_i')/(del x_i))_0 = 0` ==>`(del bar u_i)/(del x_i) = 0`
but `(del u_i)/(del x_i) =0 = ubrace((del bar u_i)/(del x_i))_(0, just shown) + (del u_i')/(del x_i) => (del u_i')/(del x_i) =0`
Substitue eq. (1) into the momentum equations and take `bar(())`
`(del u_i)/(del t) + u_j (del u_i)/(del x_j) = 1/rho (del tau_(ij))/(del x_j) + nu nabla^2 u_i`
`bar(del u_i)/(del t) = (del u_i)/(del t) + ubrace (bar(del u_i')/(del t))_0`; similarly `{ (bar(nu nabla^2 u_i) = nu nabla^2 bar(u_i)), (bar(del p)/(del x_i) = del/(del x_i) bar(bar p + p') = (del bar p)/(del x_i) etc.):}`
`bar(u_j (del u_i)/(del x_j)) = bar ((bar u_j + u_j') del/(del x_j) (bar u_i + u_i')) = bar u_j (del bar u_i)/(del x_j) + ubrace(u_j' (del bar u_i)/(del x_j))_0 + ubrace(u_j' (del u_i')/(del x_j))_0 + bar(u_j' (del u_i')/(del x_j))`
but from continuity we have
`bar(u_j' del/(del x_j) u_i') = del/(del x_j) bar(u_j'u_i') - bar(u_i' ubrace((del u_j')/(del x_j))_0`
and thus we finally obtain:
`(del bar u_i)/(del t) + bar u_j (del bar u_i)/(del x_j) =ubrace( -1/rho (del p)/(del x_i) + nu nabla^2 bar u_i)_(1/rho del/(del x_j) bar tau_(ij) ) - del/(del x_j) bar(u_i'u_j')`
Reynolds averaged N-S equation: `(del bar u_i)/(del t) + bar u_j (del bar u_i)/(del x_j) = 1/rho del/(del x_j) [ bar(tau_(ij)) - rho bar(u_i'u_j')]`
Reynolds stress: `tau_(R_(ij)) -= -rho bar(u_i'u_j')`
Turbulent Viscosity:
Turbulent-viscosity models are based on the turbulent-viscosity hypothesis. The turbulent viscosity field provides a most convenient closure to the Reynolds equations, which then have the same form. As the Navier stokes equations. It is unfortunare, therefore, that for many flows the accu racy of the hypothesis is poor. If the turbulent-viscosity hypothesis is accepted as an adequate approrimation, all that remains is to determine an appropriate specification of the turbulent viscosity. This can be written as the product of a velocity, and the task of specifying is generally approached.
The basic assumption of an effective-viscosity hypothesis is that the Reynolds stresses are uniquely related to the rates of strain and local scalar quantities. As this assumption is not valid for all flows, it is necessary to define the restricted class of flows for which it is valid. In addition, in order to effect closure, the number of scalar quantities that need be considered must be limited. An effective-viscosity assumption implies that the stresses are determined locally whereas the exact Reynolds-stress equations show that they may be convected by both the mean and fluctuating velocities. For an effective-viscosity assumption to be valid, these transport terms must be negligible. It is readily seen, from the transport equations, that the triple correlation (which accounts for turbulent convection) will be zero only when the Reynolds stresses and consequcntiy the rates of strain are homogeneous. Thus homogeneity of the rates of strain is a necessary condition for the effective-viscosity approach to be valid.
Molecular Viscosity:
The turbulent transfer of momentum by eddies creates internal fluid friction. It is the fundamental idea of how much we define viscosity in turbulent flow i.e., internal fluid resistance. Eddy viscosity what casues the internal friction.
Molecular viscosity is the same as viscosity. The coefficient of molecular viscosity is the same value as dymanic viscosity. Molecular viscosity is the transport of mass motion momentum solely by the random motions of the individual molecules not moving together in coherent groups. Molecular viscosity is analogous in laminar flow to eddy viscosity in turbulent flow.