ME 1404 Turbulence

Prof. M. K. SINHA Mechanical Engineering Department N I T Jamshedpur 831014 What is Turbulence?

• Turbulence is that state of fluid motion which is characterized by apparently random and chaotic three-dimensional vorticity. • When turbulence is present, it usually dominates all other flow phenomena and results in increased energy dissipation, mixing, heat transfer, and drag. • If there is no three-dimensional vorticity, there is no real turbulence. • Turbulent flows are not only time-dependent but space dependent as well. Characteristics of turbulent flows

• The laminar flow regime: • The flow is smooth and adjacent layers of fluid slide past each other in an orderly fashion. • The applied boundary conditions do not change with time and the flow is steady. Characteristics of turbulent flows

The turbulent flow regime: • The flow behaviour is random and chaotic. • The motion becomes unsteady even with constant imposed boundary conditions. • Turbulent fluctuations always have a three- dimensional spatial character. • Rotational flow structure, the so-called "turbulent eddies", with a wide range of length scales Turbulent flow regime

• The largest turbulent eddies interact with and extract energy from the mean flow by a process called "vortex stretching". The presence of mean velocity gradients in sheared flows distorts the rotational turbulent eddies. • Large eddies are dominated by inertia effects and viscous effects are negligible. Angular is conserved during vortex stretching and causes the rotation rate to increase and the radius of their cross-sections to decrease. Turbulent flow regime • Smaller eddies are stretched by larger eddies and weakly by the mean flow. The kinetic energy is handed down from large eddies to progressively smaller and smaller eddies in what is termed the "energy cascade". • The smallest scale of motion is dictated by . • The Reynolds number of the smallest eddies is equal to 1. Viscous effect is important. Transition from laminar to turbulent flow

• The transition to turbulence is strongly affected by factors such as pressure gradient, disturbance levels, wall roughness and heat transfer. • For engineering purposes, the major case where the transition process influences a sizeable fraction of the flow is that external wall flows at intermediate Reynolds numbers (e.g., turbomachines, helicopter rotors and low speed aircraft wings). Reynolds equations

• The instantaneous value of any flow variable can be decomposed into mean + fluctuation. • Mean and fluctuating part are denoted by an upper case and a prime ( ‘) respectively: (t) =  +(t) • Define the mean Φ of a flow property ϕ as: 1 t  = (t)dt t 0 , Reynolds equation

• The time average of the fluctuation φ΄ is zero:

1 t  = (t)dt  0 t 0 • The mathematical rules govern (t) =  +(t)  (t) =  + (t) And a fluctuating vector quantity a(t) = A + a(t) Reynolds equation

 =   = 0  + =  +   =   =  +    =  =  s s  = 0 ds = ds div a = div A div(a) = div (a) = div (A)+ div (a)

div grad  = div grad  Reynolds equation • From root-mean-square value, the fluctuations: 2 1 t rms = () = (t)dt  0 t 0 • For the rms velocity calculation (easy to measure), the kinetic energy per unit mass associated with the turbulence is: 1 k = (u2 + v2 + w2 ) 2 • And the turbulent intensity (or, the granular 1 temperature) is:  2  2  k   3  Ti = U ref Reynolds equation

• Consider the instantaneous continuity and Navier-Stoke equations for an incompressible flow with constant viscosity: div u = 0 u 1 p + div(uu) = − +  div( grad u) t  x v 1 p + div(vu) = − +  div( grad v) t  y w 1 p + div(wu) = − +  div( grad w) t  z Reynolds equation • Replacing by the sum of a mean and fluctuating component, then the time average: div u = div U = 0

U 1 P + div(UU)+ div(uu)= − +  div( grad U ) t  x V 1 P + div(VU)+ div(vu)= − +  div( grad V ) t  y W 1 P + div(WU)+ div(wu)= − +  div( grad W ) t  z Reynolds equation

• The process of time averaging has introduced new terms in the resulting time-averaged momentum equation. To reflect their roles as additional turbulent stresses on the mean velocity components (i.e. the so-called Reynolds equations):

U 1 P  uu uv uw + div(UU) = − +  div( grad U )+ − − −  t  x  x y z 

V 1 P  uv vv vw + div(VU) = − +  div( grad V )+ − − −  t  y  x y z  W 1 P  wu wv ww + div(WU) = − +  div( grad W )+ − − −  t  z  x y z  Reynolds Stresses

• The extra terms result from six additional stresses and these extra turbulent stresses are termed the Reynolds stresses:

2  = −u 2 2 xx  yy = −v  zz = −w

   xy = yx = −uv  xz = zx = −u w

 yz = zy = −vw Phenomenological theories of Turbulence • Boussinesq’s theory: • The classical models are based on the presumption that there exists an analogy between the action of viscous stresses and Reynolds stresses on the mean flow. • In Newton's low of viscosity, the viscous stresses are taken to be proportional to the rate deformation of fluid elements. For an incompressible fluid:  u u   = e =  i + j  ij ij    x j xi  Boussinesq’s theory • An analogy: Reynolds stresses could be linked to mean rate of deformation :  U U   = −uu =   i + j  ij i j t    x j xi  • where μt is the turbulent viscosity. • By analogy, turbulent transport of a scalar is taken to be proportional to the gradient of the mean value of the transported quantity:        − ui = t   • where Γt is the turbulent diffusivity.  xi  t • Schmidt number:  t = t Prandtl’s mixing length theory • Assuming the kinetic turbulent viscosity can be expressed as a product of a turbulent velocity scale (m/s) and a length scale (m): vt = C l

• and the dynamic turbulent viscosity is: t = C l • Most of the kinetic energy of turbulent is contained in the largest eddies and the turbulence length scale l is therefore characteristic of these eddies which interact with the mean flow. • Link the characteristic velocity scale of the eddies with the mean flow properties: Prandtl’s mixing length • In simple two-dimensional turbulent flows, the only significant is

 xy = yx = −uv U • The only significant mean velocity gradient is y U  = cl y

The Prandtl’s mixing length model: U v =l 2 t m y Von-Karaman mixing length

• The turbulent Reynolds stress is: U U  = = −uv =  l 2 xy yx m y y • Since turbulence is a function of the flow, lm is also a function of the flow. • Von-Karaman extended the Prandtl’s mixing length theory using similarity analysis for estimating mixing length as Applications of mixing length

• The mixing length model gives good prediction in flows where the turbulent properties develop in proportion to a mean flow length scale. • Its universal popularity in calculations of flows around wing section. • Mixing length models are the most widely used turbulent models in external aerodynamics calculations in the aerospace industry. Universal distributions near a solid wall

• At high Re, if y is the co-ordinate direction normal

to the wall, the mean velocity at a point yp with + 30 < yp < 500 satisfies the log-law: U 1 u 2 3 u + = = ln(Ey + ) k =  u p C  = u   y  u y u = w y + =  • Where friction velocity    • Von Karman’s constant κ = 0.41 and the wall roughness parameter E = 0.8 for smooth walls. • At low Re, the log-law is not valid. THANK YOU