Date post: | 04-Jul-2015 |
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Engineering |
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Boundary Layer Equations
Prof. Rohit GoyalProfessor, Department of Civil Engineering
Malaviya National Institute of Technology Jaipur
E-Mail: [email protected]
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Topics Covered
Different Boundary Layer Thickness Nominal Thickness Displacement Thickness Momentum Thickness Energy Thickness
Equations for different BL thickness Boundary Layer Equations
Assumptions
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Nominal Thickness (δ)
Nominal thickness of the boundary layer is defined as the thickness of zone extending from solid boundary to a point where velocity is 99% of the free stream velocity (U)
This is arbitrary, especially because transition from 0 velocity at boundary to the U outside the boundary takes place asymptotically.
It is based on the fact that beyond this boundary, effect of viscous stresses can be neglected.
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Other Definitions of BL Thickness
Many other definitions of boundary layer thickness has been introduced at different times and provide important concepts based on mathematical calculations and logic
These definitions are Displacement Thickness (δ*) Momentum Thickness (θ) Energy Thickness (δe)
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Displacement Thickness
Presence of boundary layer introduces a retardation to the free stream velocity in the neighborhood of the boundary
This causes a decrease in mass flow rate due to presence of boundary layer
A “velocity defect” of (U-u) exists at a distance y along y axis
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Displacement Thickness
Displacement thickness may be thought of as the distance (measured perpendicular to the boundary) with which the boundary may be imagined to have been shifted such that the actual flow rate would be the same as that of an ideal fluid (with slip) flowing around the displaced boundary
This may be imagined in as explained in figures on next page
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Velocity Distribution
U
SolidBoundary
Equivalent Flow Rate
U
Velocity Defect
VelocityDefect
δ *
Ideal FluidFlow
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Eqn. for Displacement Thickness
By equating the flow rate for velocity defect to flow rate for ideal fluid
If density is constant, this simplifies to
δ* would always be smaller than δ
( )∫ −=δ
ρδρ0
* dyuUU
∫
−=
δδ
0
* 1 dyU
u
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Momentum Thickness
Retardation of flow within boundary layer causes a reduction in the momentum flux too
So similar to displacement thickness, the momentum thickness (θ) is defined as the thickness of an imaginary layer in free stream flow which has momentum equal to the deficiency of momentum caused to actual mass flowing inside the boundary layer
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Eqn. for Momentum Thickness
By equating the momentum flux rate for velocity defect to that for ideal fluid
If density is constant, this simplifies to
θ would always be smaller than δ* and δ
( ) ( )∫ −=δ
ρθρ0
2 uUudyU
∫
−=
δθ
01 dyU
u
U
u
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Graphical Representation
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Energy Thickness
Similarly Energy thickness (δe) is defined as the thickness of an imaginary layer in free stream flow which has energy equal to the deficiency of energy caused to actual mass flowing inside the boundary layer
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Eqn. for Energy Thickness
By equating the energy transport rate for velocity defect to that for ideal fluid
If density is constant, this simplifies to
( ) ( )∫ −=δ
ρδρ0
222
2
1
2
1uUudyU e
∫
−=
δδ
0 2
2
1 dyU
u
U
ue
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Boundary Layer Assumptions
Following assumptions are made for the analysis of the boundary layer It is assumed (also observed to great extend) that
Reynolds number of flows are large and the thickness of boundary layer are small in comparison with any characteristic dimension of the boundary
The boundary is streamlined so that the flow pattern and pressures determined by ideal flow theory are accurate
It is possible to treat the flow at constant density and isothermal conditions prevail so that viscosity is also constant
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Approximations Made
Using these assumptions following approximations are made The pressure does not vary across any given
section of the boundary layer. So pressure determined by ideal fluid theory at the edge of boundary holds within the boundary layer also
Since flow is essentially parallel so that the shear stress are solely determined by Newton’s law of viscosity τ=µ(∂u/∂y)
Compared with thin boundary layer, the gentle curvature of the boundary has practically no influence on the flow properties
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Coordinate System
Last approximations allows us to choose coordinate system with x-axis along the curved body and y axis along normal to boundary as shown below
Strictly speaking this coordinate system is curvilinear but is expected to behave like a rectangular system in the thin region of the boundary layer
Continuity Equation
Only steady two dimensional flow is considered for simplicity
Continuity Equation in 2D is ∂u/∂x+∂v/∂y = 0
Where u and v are velocity components in x and y axes
Momentum Equation
Since velocity component in y direction is negligibly small so momentum equation is considered only in x direction
Considering a small control volume of sides ∆x and ∆y and thickness in the third direction as unity is considered as shown on next page.
Control Volume
Summation of Forces
Neglecting component of gravity in x-direction, only pressure and shear stress as shown on control volume are considered
There would be a negative shear stress on lower face because layer below is trying to retard the motion of particles within control volume. Similarly shear stress on top surface would be positive
xyy
xyxx
ppypFx ∆
∆
∂∂++∆−∆
∆
∂∂+−∆=∑ τττ
Forces in x-Direction
So total force in x direction
Using Newton’s law of viscosity
This would be equal to change in rate of momentum in x-direction
xyy
xyxx
ppypFx ∆
∆
∂∂++∆−∆
∆
∂∂+−∆=∑ τττ
yxy
u
x
pFx ∆∆
∂∂+
∂∂−=∑ 2
2
µ
Change in rate of momentum
From Left and Right Faces
Mass entering the left face = ρu∆y Momentum entering the left face =
ρu2∆y Momentum leaving right face
= ρu2∆y + ∂(ρu2∆y)/∂x ∆x = ρ(u2+ ∂u2/∂x ∆x)∆y
From Top and Bottom faces
Mass entering from bottom face = ρv∆x Momentum entering the bottom face =
(ρv∆x)u Momentum leaving top face
= ρuv∆x + ∂(ρuv∆x)/∂y ∆y = ρ(uv+ ∂(uv)/∂y ∆y)∆x
Net momentum in x-direction
( )
xuvyuxyy
uvuvyx
x
uu ∆−∆−∆
∆
∂∂++∆
∆
∂∂+ ρρρρ 2
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Net Rate of Momentum
Net momentum in x-direction simplifies to
Using continuity equation ∂v/∂y=-∂u/∂x So the net rate of momentum
yxy
uv
y
vu
x
uu ∆∆
∂∂+
∂∂+
∂∂
2ρ
yxy
uv
x
uu ∆∆
∂∂+
∂∂ρ
Prandtl BL Equation
Equating net rate of momentum to forces in x-direction
Which simplifies to
This is also referred as Prandtl BL Eqn.
yxy
u
x
pyx
y
uv
x
uu ∆∆
∂∂+
∂∂−=∆∆
∂∂+
∂∂
2
2
µρ
2
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y
u
x
p
y
uv
x
uu
∂∂+
∂∂−=
∂∂+
∂∂ υ
ρ