Fluid Mechanics : Control-Volume Analysis , Laminar Flow & Turbulent Flow Notes

By Deepanshu Rastogi|Updated : November 27th, 2021

Complete coverage of syllabus is a very important aspect for any competitive examination but before that important subject and their concept must be covered thoroughly. In this article, we are going to discuss the fundamental of Control-Volume Analysis, Laminar Flow & Turbulent Flow which is very useful for AE JE Exams.   

 

Control-Volume Analysis & Laminar Flow & Turbulent Flow

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

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  • From conservation of mass,

mass into face 1 = mass out of face 2

  • The rate at which momentum enters face 1 is

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  • The rate at which momentum leaves face 2 is

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

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  • The Momentum equation is: F = m(u2 - u1)

 

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

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

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  •  And the force in the y-direction:

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

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And the angle of this force: 

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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

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  • 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,

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  • In the x-direction: 

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  • In the y-direction: 

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Using Bernoulli Equations here

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  • 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

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

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  • 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

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  • And the resultant force on the fluid is given by 

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  • And the direction of application is 

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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

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  • In the x-direction

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  • 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

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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.

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  •  Total force in the x direction

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  • In the y-direction,

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  • 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 

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  • And the resultant force on the fluid is given by

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  •  And the direction of application is 

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

 

 

Laminar Flow

Laminar Flow

Laminar Flow

  • Laminar flow is also known as viscous flow
  • In laminar flow, viscous force is highly is highly predominant.

Case–I

Laminar flow between 2- parallel plates

Consider a fluid element in the flow field. An element has thickness dy, length dx and y distance from bottom plate.

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Assumption:- width of flow perpendicular to paper = unity

Free body diagram of element

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Apply equilibrium condition

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But we know that byjusexamprep

So, byjusexamprep

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By integrating equation (1)

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⇒ Again integrate with respect to y

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Apply boundary condition

  • No slip condition
  • Due to friction velocity of fluid at upper and bottom plate is zero.
  1. At y = 0, u = 0
    C2 = 0
  2. At y=H, u=0
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So, byjusexamprep

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Maximum velocity

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Mean velocity

Mass flow rate = image014

Mass flow rate, when consider average velocity.

image015

By equating both term {from eq. (a) and (b)} and putting expression of image016

image017

 

From expression of image018

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Shear stress distribution:

By Newton’s Law of viscosity

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Pressure difference b/w two points along flow

Consider average velocity expression

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Laminar flow through pipe: (circular)

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Consider a fluid element having radius r and length dx

Free body diagram of element

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Apply horizontal equilibrium equation

Internal flow:-

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According to Newton’s Law of Viscosity

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from first figure in this section

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put the value of image029 in eq. (a)

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By integrating it

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At image032 ---no slip condition

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Maximum Velocity:

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So from expression of u, put r=0

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Mean Velocity:

Mass flow rate is constant throughout the pipe

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From the expression of image018

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Pressure distribution:

In calculation of pressure difference always consider average velocity So.

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By Rearranging

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Velocity and shear stress profile in circular pipe

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Head Loss

  • Flow between Flat plate
    We know that image043
    Head Loss 
    image044
  • Circular pipe flow
    We know that image045
    Head Loss image046
    image047

NOTE:

  • Dependency of flow on Reynolds number
    Reynolds number image048
    image049 density of fluid
    image050 Velocity of flow
    image051 Dynamic Viscosity
    image052 Characteristic dimension
    For Laminar flow image053
    If, image054 Turbulent flow
    If, 2000 image055 transition flow
  • Head Loss equation by Darcy’s
    image056
    Here image057 Friction factor
    image058 Velocity (average)
    image059 Length of pipe
    image052 Diameter of pipe
    This equation is valid for both turbulent and laminar flow.
  • Friction factor for circular pipe flow
    In circular pipe
    image060
    But Darey’s equation
    image061

By equating both expressions

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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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