FLUID MECHANICS :laminar and turbulent flow Notes

By Deepanshu Rastogi|Updated : April 1st, 2021



Flow Through Pipes & Control-Volume Analysis

Control-Volume Analysis

Control-Volume Analysis of Mass, Momentum and Energy is an important topic of Fluid mechanics which deals with topics such as control mass, control volume, momentum equation, continuity equation and Impact of Jets on planes and vanes. 

Control Mass

  • A fixed mass of a fluid element in the flow-field is identified and conservation equations for properties such as momentum, energy or concentration are written.
  • The identified mass moves around in the flow-field.
  • Its property corresponds to the same contents of the identified fluid element may change from one location to another.

Control Volume

  • This approach is popular and widely applied in the analysis.
  • An arbitrarily fixed volume located at a certain place in the flow-field is identified and the conservation equations are written.
  • The property under consideration or analysis may change with time.

The Momentum Equation 

  • It relates the sum of the forces to the acceleration or rate of change of momentum


  • From conservation of mass,

mass into face 1 = mass out of face 2

  • The rate at which momentum enters face 1 is


  • The rate at which momentum leaves face 2 is


  • Thus the rate at which momentum changes across the stream tube is 


  • The Momentum equation is:



  • This force acts on the fluid in the direction of the flow of the fluid
  • If the Motion is not one Dimensional


  • We consider the forces by resolving in the directions of the co-ordinate axes.
  •  The force in the x-direction



  •  And the force in the y-direction:


  • The resultant force can be found by combining these components :


And the angle of this force: 


Application of the Momentum Equation

Force due to the flow around a pipe bend

  • A converging pipe bend lying in the horizontal plane turning through an angle of Θ


  • As the fluid changes direction a force will act on the bend.
  • This force can be very large in the case of water supply pipes. The bend must be held in place to prevent breakage at the joints

Taking Control Volume,


  • In the x-direction: 


  • In the y-direction: 


Using Bernoulli Equations here



  • where hf is the friction loss (this can often be ignored, hf=0)
  • As the pipe is in the horizontal plane, z1=z2 And with continuity, Q= u1A1 = u2A


  • Knowing the pressures at each end the pressure force can be calculated, 


  • There are no body forces in the x or y directions, so FRx = FRy = 0
  • The body force due to gravity is acting in the z-direction so need not be considered



  • And the resultant force on the fluid is given by 


  • And the direction of application is 



Impact of a Jet on a Plane

A jet hitting a flat plate (a plane) at an angle of 90º

  • The reaction force of the plate. i.e. the force the plate will have to apply to stay in the same position


  • In the x-direction


  • The system is symmetrical the forces in the y-direction cancel, so, F= 0
  • The pressures at both the inlet and the outlets to the control volume are atmospheric. The pressure force is zero, 

Fpx= Fpy= 0

  • As the control volume is small we can ignore the body force due to gravity, FBx= FBy= 0


Force on a curved vane

  • Pressures at ends are equal which is equal atmospheric pressure.
  • Both the cross-section and velocities (in the direction of flow) remain constant.


  •  Total force in the x direction




  • In the y-direction,


  • The pressure at both the inlet and the outlets to the control volume is atmospheric, Fpx=Fpy=0
  • No body forces in the x-direction, FBx=0
  • In the y-direction, the body force acting is the weight of the fluid.
  • If V is the volume of the fluid on the vane then, FBX= ρgV.

The resultant force in the 



  • And the resultant force on the fluid is given by


  •  And the direction of application is 


The force on the vane is the same magnitude but in the opposite direction, R= -FR.




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Turbulent Flow

Turbulent Flow: Turbulent flow is a flow regime characterized by the following points as given below

Shear stress in the turbulent flow

12-Elementary-turbulent-flow (1)


where, τv and τ= shear stress due to viscosity and turbulence.

η = eddy viscosity coefficient.

Turbulent shear stress by Reynolds

τ = ρu’v’

u' and v’ fluctuating component of velocity.

Shear stress in turbulent flow by Prandtl

12-Elementary-turbulent-flow (2)


where, l = Mixing length

The velocity distribution in the turbulent flow for pipes is given by the expression.

12-Elementary-turbulent-flow (3)

Umax = centre velocity

where, y = Distance from the pipe wall, R = radius of the pipe

u* = Shear velocity 12-Elementary-turbulent-flow (4)

Velocity defect is the difference between the maximum velocity (umax) and local velocity (u) at any point is given by

12-Elementary-turbulent-flow (5)

Karman-Prandtl velocity distribution equation.

Hydro dynamically pipe

12-Elementary-turbulent-flow (6)


12-Elementary-turbulent-flow (7)

where, u = velocity at any point in the turbulent flow

u* = shear velocity = 12-Elementary-turbulent-flow (8)

v = Kinematic viscosity of fluid

y = Distance from pipe wall

k = Roughness factor

Velocity distribution in terms of average velocity

12-Elementary-turbulent-flow (9)


12-Elementary-turbulent-flow (10)

Common Mean Velocity Distribution Equation: (Very Important for Exams)

[(u-u¯)/ u*]= 5.75 log10 (y/R) + 3.75             

(This is valid for both rough and smooth pipes, that's why it is referred as Common Mean Velocity Distribution Equation)


Coefficient of friction 12-Elementary-turbulent-flow (12) (for laminar flow)

12-Elementary-turbulent-flow (13) (for smooth pipe)

12-Elementary-turbulent-flow (14)



(for smooth pipe)

12-Elementary-turbulent-flow (15) (for rough pipe)



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