Fluid Mechanics Applications/B19: VISCOUS FLOW

Introduction

Practical and real world applications of the fluid mechanics deals with viscous flow and turbulence. While viscosity consideration is the major aspect of viscous flows, the essential feature of turbulent flows is that the flow field varies significantly and irregularly in both space and time. Since the flow field varies with both space and time, statistical approaches are adopted to characterise such flows. One such widely adopted method for Navier Stokes Equations is resorting to average the flow field over time, or sometimes commonly known as Reynolds Averaged Navier Stokes(RANS) equations. As is the practise for simple fluid flows, assuming incompressible flow conditions, i.e. constant phase density and viscosity with no thermal interaction, the continuity and momentum equations become:

 Continuity: ${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}$ Momentum: ${\displaystyle \rho {\frac {dV}{dt}}=-\nabla p+\rho g+\mu \nabla ^{2}V}$

Reynold’s Time-Averaging concept

For turbulent flow because of the fluctuations, every velocity and pressure term in the continuity equation is a rapidly varying random function of time and space. We can't solve random functions ${\displaystyle V(x,y,z,t)}$ and ${\displaystyle p(x,y,z,t)}$ for a single pair. So we take mean or average values of velocity, pressure, shear stress, and the like in a high Reynolds-number (turbulent flow). The time mean ${\displaystyle {\overline {u}}}$ of a turbulent function ${\displaystyle u(x,y,z,t)}$ is defined by

${\displaystyle {\overline {u}}={\frac {1}{T}}\int _{0}^{T}u\,dt}$

In the above equation ${\displaystyle T}$ is the averaging period taken to be longer than any significant period of the fluctuations themselves. The mean value of turbulent velocity is shown in figure given below. for turbulent gas and water flows, an averaging ${\displaystyle T=5}$ sec is usually quite adequate.

In the above graph the fluctuation of u from its average value is defined as

${\displaystyle {\acute {u}}=u-{\overline {u}}}$

It follows by definition that a fluctuation has zero mean value:

${\displaystyle {\overline {u}}'={\frac {1}{T}}\int _{0}^{T}(u-{\overline {u}})\,dt={\overline {u}}-{\overline {u}}=0}$

Although, the mean square of a fluctuation is not zero and is a measure of the intensity of the turbulence:

${\displaystyle {\overline {{u}'^{2}}}={\frac {1}{T}}\int _{0}^{T}{\overline {u'}}^{2}\,dt\neq 0}$

Reynolds’s idea was to split each property mean plus fluctuating variables:

• ${\displaystyle u=U_{avg}+u'}$
• ${\displaystyle v=V_{avg}+v'}$
• ${\displaystyle w=W_{avg}+w'}$
• ${\displaystyle p=P_{avg}+p'}$

Substituting these values in the continuity equation we get

${\displaystyle {\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}+{\frac {\partial W}{\partial z}}=0}$

Where

• ${\displaystyle U=U_{avg}}$ (average value of ${\displaystyle u}$)
• ${\displaystyle V=V_{avg}}$ (average value of ${\displaystyle v}$)
• ${\displaystyle W=W_{avg}}$ (average value of ${\displaystyle w}$)

Therefore, above equation is no different from laminar continuity relation. However, each component of the momentum equation, after time averaging, will contain mean values plus three mean products, or correlation, of fluctuating velocities. So the momentum equation is reduced to

${\displaystyle \rho {\frac {\partial U}{\partial t}}=-{\frac {\partial P}{\partial x}}+\rho g+{\frac {\partial }{\partial x}}(\mu {\frac {\partial U}{\partial x}}-\rho U\cdot U)+{\frac {\partial }{\partial y}}(\mu {\frac {\partial U}{\partial y}}-\rho U'V')+{\frac {\partial }{\partial z}}(\mu {\frac {\partial U}{\partial z}}-\rho U'W')}$

The three correlations ${\displaystyle -\rho U\cdot U,-\rho U'V',-\rho U'W'}$ are called "turbulent stresses" because they have the same dimensions and occur right alongside the Newtonian (laminar) stress terms ${\displaystyle \mu {\frac {\partial U}{\partial x}}}$ and so on. The figure given below shows the velocity and shear distribution

The Logarithmic Overlap Law

There are three regions in turbulent flow near a wall:

1. Wall layer: Viscous shear dominates.
2. Outer layer: Turbulent shear dominates.
3. Overlap layer: Both types of shear are important.

From now on let us agree to drop the over bar from velocity ${\displaystyle {\overline {u}}}$. Let ${\displaystyle \tau w}$ be the wall shear stress, and let ${\displaystyle \delta }$ and ${\displaystyle U}$ represent the thickness and velocity at the edge of the outer layer,

${\displaystyle y=\delta }$

For the wall layer, Prandtl deduced in 1930 that ${\displaystyle u}$ must be independent of the shear layer thickness ${\displaystyle u=f(\mu ,\tau w,\rho ,y)}$

By dimensional analysis, this is equivalent to

u +=u/u*=F(yu*/v)
u*=(τw/ρ)1/2

Above equation is called the law of the wall, and the quantity u* is termed the friction velocity because it has dimensions {LT-1}, although it is not actually a flow velocity. Subsequently, Kármán in 1933 deduced that u in the outer layer is independent of molecular viscosity, but its deviation from the stream velocity U must depend on the layer thickness ${\displaystyle delta}$ and the other properties:

(U-u)outer =g(δ,τw,ρ,y)

Again, by dimensional analysis we rewrite this as

(U-u)/u*=G(y/δ)

where u* has the same meaning. This equation is called the velocity-defect law for the outer layer. Both the wall law and the defect law are found to be accurate for a wide variety of experimental turbulent duct and boundary-layer flows [1 to 3]. They are different in form, yet they must overlap smoothly in the intermediate layer. In 1937 C. B. Millikan showed that this can be true only if the overlap-layer velocity varies logarithmically with y:

u*/u = (1/K)ln(yu*/v) +B
OVERLAP LAYER

Over the full range of turbulent smooth wall flows, the dimensionless constants K and B are found to have the approximate values K= 0.41 and B=5.0 Above equation is called the logarithmic-overlap layer.

Thus by dimensional reasoning and physical insight we infer that a plot of u versus ln y in a turbulent-shear layer will show a curved wall region, a curved outer region, and a straight-line logarithmic overlap. Figure given below shows that this is exactly the case.

The four outer-law profiles shown all merge smoothly with the logarithmic overlap law but have different magnitudes because they vary in external pressure gradient .the wall law follows the linear viscous relation

u+ =u/u* = yu*/v = y+

from the wall to about y+ =5, therefore curving over to merge with the logarithmic law at about y+ =30.

Turbulence modeling is a very active field. Now a research available, which is confined to the use of the logarithmic law for pipe and boundary layer problems.

-ρU'V' =τ(turbulent) =μdu/dy
where
μ =ρL²du/dy

Where μ is the property of the flow, not the fluid, is called the eddy viscosity and can be modeled in various ways and L is called the mixing length of the turbulent eddies. Near a solid wall, L is approximately proportional to distance from the wall, according to Karman:

L = ky
where
k = Karman’s constant = 0.41

Conclusion

We have seen that in the turbulence flow, if we take the time averaging values of velocity and pressure in the continuity and momentum equation than the reduced continuity equation is same but reduced momentum equation is changed causes turbulent stresses developed in the flow, in which laminar shear is dominated near the wall and turbulent shear dominates in the outer layer, the intermediate region is called overlap layer. We have also seen that the experimentally verification of the inner, outer and overlap layer laws relating velocity profile in turbulence wall flow in which actually approximates nearly the entire velocity profile, except for the outer law when the pressure is increasing strongly downstream. Modern turbulence models approximate three-dimensional turbulent flows and employ additional partial differential equations for such quantities as the turbulence kinetic energy, the turbulent dissipation, and the six Reynolds stresses.

References

• Fluid Mechanics(SI UNIT)-Seventh edition, by FRANK M WHITE.
• Engineering Fluid Mechanics, by Prof. K. L. KUMAR
• INTERNET