Module-1 Turbulent Flow€¦ · Module-1 Turbulent Flow 10 Darshan Institute of Engineering & Technology, Rajkot Ratio of inertia force of a flowing fluid and viscous force of the

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Module-1

Turbulent Flow

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98795 10743

Applied Fluid Mechanics (2160602)

19-02-2020

Prof. Mehul Pujara

9Module-1 Turbulent Flow Darshan Institute of Engineering & Technology, Rajkot

Ratio of inertia force of a flowing fluid and viscous force of the fluid.

Re = Fi / Fv

Inertia force = Mass * Acceleration of flowing fluid

= Density * Volume * Velocity/time

= ρ * Q * v

= ρ * A * v * v

= ρ * A * v2

Viscous force = Shear stress * Area (τ = μ du/dy)

= μ du/dy * A

= μ v/L * A

1. Reynold Number:


Ratio of inertia force of a flowing fluid and viscous force of the fluid.

Re = Fi / Fv

=ρAv2

μv

LA

=ρvL

μ

Higher the Re, greater the inertia effect. Smaller the Re, greater the viscouseffect.

Examples of such situation:

I. Flow of incompressible fluid in a pipe

II. Motion of submarine

1. Reynold Number:


The shear stress in viscous flow is given by Newton’s law of viscosity as

τ𝑣 = μ𝑑𝑢

𝑑𝑦, τ𝑣 = shear stress due to viscosity

Similarly J. Boussinesq has given turbulent shear stress in mathematicalform as

τ𝑡 = η𝑑 𝑢

𝑑𝑦, τ𝑡 = shear stress due to turbulence

η = eddy viscosity

The ratio of η and ρ is known as kinematic eddy viscosity.

ε =ηρ

Shear Stress in Turbulent Flow:


Reynold in 1866 has suggested the turbulent shear stress as

τ = ρ 𝑢 𝑣,

𝑢 𝑣 = fluctuating component of velocity in the direction of x and y due toturbulence

The average shear stress is given by

τ = ρ𝑢𝑣

The above turbulent shear stress is known as Renold stress.

Shear Stress in Turbulent Flow:


Turbulent stress can be calculated from the value of 𝑢 𝑣 is known.

τ = ρ 𝑢 𝑣,

But it is very difficult to measure 𝑢𝑣

τ = ρ𝑢𝑣

The above difficulty is solved by L. Prandtl in 1925 by introducing mixinglength hypothesis.

The mixing length l, is that distance between two layers in the transversedirection such that the lumps of fluid particles from one layer could reachthe other layer and the particles are mixed in the other layer in such a waythat the momentum of the particles in the direction of x is same.

The value of 𝑢 𝑣 is given as

𝑢 = l𝑑𝑢

𝑑𝑦

𝑣 = l𝑑𝑢

𝑑𝑦

Prandtl Mixing Length Theory:


The value of 𝑢𝑣 is given as

𝑢𝑣 = l𝑑𝑢

𝑑𝑦* l

𝑑𝑢

𝑑𝑦

= l2 (𝑑𝑢

𝑑𝑦)2

The value of turbulent stress is

τ = ρ l2(𝑑𝑢

𝑑𝑦)2

The total shear stress at any point in turbulent flow is given by

τ or τ = μ𝑑𝑢

𝑑𝑦+ ρ l2(

𝑑𝑢

𝑑𝑦)2

Prandtl Mixing Length Theory:


In case of turbulent flow, the total shear stress at any point is thesum of viscous shear stress and turbulent shear stress.

Also the viscous shear stress is negligible except near the boundary.So shear stress is calculated using equation,

τ or τ = ρ l2(𝑑𝑢

𝑑𝑦)2

From this equation, the velocity distribution can be obtained if therelation between l, the mixing length and y is known.

Prandtl assumed that the mixing length, I is a linear function of thedistance y from the pipe wall i.e., I = ky, where k is a constant, knownas Karman constant and = 0.4

τ or τ = ρ (ky)2 (𝑑𝑢

𝑑𝑦)2

τ or τ = ρ k2y2 (𝑑𝑢

𝑑𝑦)2

Velocity distribution in turbulent flow:


Rewriting the below equation,

τ or τ = ρ k2y2 (𝑑𝑢

𝑑𝑦)2

(𝑑𝑢

𝑑𝑦)2 = τ / ρ k2y2

𝑑𝑢

𝑑𝑦= τ / ρ k2y2

𝑑𝑢

𝑑𝑦=

1

𝑘𝑦

τ

ρ

For small values of y that is very close to the boundary of the pipe,Prandtl assumed shear stress τ to be constant and approximatelyequal to τ0 which presents the turbulent shear stress at the pipeboundary. Substituting τ = τ0 in above equation

𝑑𝑢

𝑑𝑦=

1

𝑘𝑦

τ0ρ



𝑑𝑢

𝑑𝑦=

1

𝑘𝑦

τ0ρ

In above equation the dimension of termτ0ρ

is𝐿

𝑇which is velocity

dimension. So,

𝝉0ρ

= shear velocity = u*

So,𝑑𝑢

𝑑𝑦=

1

𝑘𝑦u*

Integrating above equation,

u =u∗𝑘

ln y + C



Equation shows that in turbulent flow, the velocity varies directly withthe logarithm of the distance from the boundary.

In other words the velocity distribution in turbulent flow islogarithmic in nature.

To determine the constant of integration, C the boundary conditionthat at y = R (radius of pipe), u = umax is substituted in equation

umax =u∗𝑘

ln R + C

∴ C = umax -u∗𝑘

ln R

Putting the value of C in equation u =u∗𝑘

ln y + C

u =u∗𝑘

ln y + umax -u∗𝑘

ln R

= umax +u∗𝑘

ln (y/R)



u =u∗𝑘

ln y + umax -u∗𝑘

ln R

= umax +u∗𝑘

ln (y/R)

Putting the value of k= 0.4 = Karman constant

u = umax +u∗0.4

ln (y/R)

u = umax + 2.5 u∗ ln (y/R)

Equation is called ‘PrandtI’s universal velocity distribution equationfor turbulent flow in pipes. This equation is applicable to smooth aswell as rough pipe boundaries.



Let k is the average height of theirregularities projecting from thesurface of a boundary as shown inFigure

If the value of k is large for a boundarythen the boundary is called roughboundary

if the value of k is less, then boundaryis known as smooth boundary.

in general. This is the classification ofrough and smooth boundary based onboundary characteristics. But for properclassification, the flow and fluidcharacteristics are also to beconsidered.

Hydrodynamically Smooth and Rough Boundaries :


For turbulent flow analysis along aboundary, the flow is divided in twoportions.

The first portion consists o f a thinlayer of fluid in the immediateneighbourhood of the boundary, whereviscous shear stress predominateswhile the shear stress due to turbulenceis negligible. This portion is known aslaminar sub-layer.

The second portion of flow, whereshear stress due to turbulence are largeas compared to viscous stress is knownas turbulent zone.

Hydrodynamically Smooth and Rough Boundaries :


References:

1. Fluid Mechanics and Fluid Power Engineering by D.S. Kumar, S.K.Kataria & Sons

2. Fluid Mechanics and Hydraulic Machines by R.K. Bansal, LaxmiPublications

3. Fluid Mechanics and Hydraulic Machines by R.K. Rajput, S.Chand & Co

4. Fluid Mechanics; Fundamentals and Applications by John. M. CimbalaYunus A. Cengel, McGraw-Hill Publication

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Module-1 Turbulent Flow€¦ · Module-1 Turbulent Flow 10 Darshan Institute of Engineering & Technology, Rajkot Ratio of inertia force of a flowing fluid and viscous force of the