review for exam 3 - university of...
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Review for Exam 3
Chapter 9: Differential analysis of Fluid Flow Governing equation of: Integral form Useful for large scale CV analysis Differential form Useful for relatively small-scale point analysis
9.1 Continuity equation in differential form
**Simplification of 2. denotes the incompressibility of the fluid!!
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9.2 The stream function ψ
Cartrsian coordinates: x
vy
u∂∂
−=∂∂
=ψψ ,
Polar coordinates: r
ur
ur ∂∂
−=∂∂
=ψ
θψ
θ ,1
Axisymmetric cylindrical coordinates: r
uzr
u zr ∂∂
−=∂∂
=ψψ ,1
Polar coordinates (left), Axisymmetric cylindrical coordinates (right)
Important features of ψ • Curves of constant ψ are streamlines of the flow • The difference in the value of ψ from one streamline to another is equal to
the volume flow rate per unit width between the two streamlines
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9.3 Navier-Stokes (NS) Equations NS equation is a conservation of momentum!!
Start from 1-D flow approximation:
Notice that: Body force due to external fields such as gravity or magnetics
Surface force due to the stresses acting on the sides of CS
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The physical meanings of each term in NS equation are:
Write viscous shear and normal stresses in the form as:
After some mathematical manipulation, NS and continuity equations are obtained as:
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9.4 Differential Analysis of Fluid Flow
• Couette Flow
Governing equations and boundary conditions for flow field:
Solution (velocity profile and shear stress):
• Generalization of inclined flow with a constant pressure gradient
Governing equations and boundary conditions for flow field:
Solution with non-dimensional form (velocity profile):
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Further non-dimensionalization of the obtained velocity profile yields:
Effect of pressure gradient to the velocity profile of Couette flow:
linear is profile velocity Thegradient pressure Zero:0 wallstationary near the Backflow gradient pressure Adverse :0
widthentire over the positive isVelocity gradient pressure Favorable :0
→=→<→>
PPP
Chapter 10: Approximate Solutions of the NS equations
10.1 Creeping flow approximation
In non-dimensional form of NS equation,
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10.2 Approximation for Inviscid Regions of Flow See Text pp.481-pp.485 10.3 The Irrotational Flow Approximation See Text pp.485-pp.510 10.4 Qualitative Description of the Boundary layer Boundary layer:
A very thin region of flow near a solid wall where viscous forces and rotationality cannot be ignored.
Boundary-layer theory:
The asymptotic form of the NS equations for high-Re flow about the slender bodies.
The order assumptions of boundary-layer theory:
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Use the order assumptions above to obtain boundary-layer equations from NS equations: NS equations:
Boundary-layer equations:
Important aspects of boundary-layer equations:
1. 0=∂∂yP i.e. P=Pe=const. across the boundary layer
2. Continuity equation holds. 3. Boundary conditions to solve the boundary-layer equations are:
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10.5 Quantitative relations for the Laminar Boundary Layer Laminar boundary layer over a flat plate Blasius solution Governing equations and boundary conditions to obtain Blasius solution:
Results:
Displacement thickness: imaginary increase in thickness of the wall, as seen by the outer flow, due to the effect of the growing boundary layer.
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Momentum thickness: the loss of momentum flux per unit width divided by ρU2 due to the presence of the growing boundary layer.
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10.6 Qualitative relations for the Turbulent Boundary layer Transition from laminar boundary-layer to turbulent boundary-layer:
Engineering critical Reynolds number: Rex, cr= 5105× Rex<Rex, cr: boundary layer is most likely laminar. Rex>Rex, cr: boundary layer is most likely turbulent. Description of turbulent flow:
Velocity and pressure are random functions of time!! They can be separated into two parts such as mean and fluctuation components:
Most important influence of turbulence on the mean motion:
An increase in the fluid stress by “Reynolds stresses”
‘Modeling’ required!!
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Mean flow equations for turbulent flow Reynolds Averaged Navier Stokes (RANS) equations
Turbulence Modeling: a) Eddy viscosity, Mixing-length theory, One-equation model,
Two- equation model (k-ε model, k-ω model: Recall CFD-PreLab2, Lab2!!)
b) Mean-flow velocity profile correlations
1) Inner layer
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2) Outer layer
3) Overlap layer
Momentum Integral Analysis: obtain general momentum integral relation Valid for both laminar and turbulent flow
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On the other hand, CD can be expressed as;
Then,
Therefore,
Finally, momentum integral relation is obtained as:;
Approximate solutions for a laminar boundary layer obtained from momentum integral analysis:
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Approximate solutions for a turbulent boundary layer obtained from momentum integral analysis: Velocity profile inside the turbulent boundary layer to obtain the solutions: 1) log-law
2) 1/7 power law
Obtained approximate solutions:
Cf vs ReL relationship in laminar and turbulent boundary layer
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Chapter 11: Drag and Lift 11.1 Basic consideration
[Drag]=[form drag]+[skin-friction drag]
Drag
Lift
Drag reduction by streamlining:
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11.2 Drag of 2-D and 3-D bodies Flat-plate parallel to the flow:
In general,
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Drag coefficients if common geometries (2D and 3D):
Flow over cylinder and spheres:
• Re<1: Creeping flow, CD=24/Re, No-flow separation regime • 105<Re<106: boundary layer become turbulent, large reduction in CD
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Lift:
ACWVL max,
min2
:ocityflight vel Minimum
ρ=
Lift generation by spinning: Magnus effect