1. Boundary Layer Theory (Analysis)
Dept. of Naval Architecture and Ocean Engineering

2. Contents Importance of Viscous Phenomena
Conditions of at a Fluid-Solid Boundary (Concept of Continuum Mechanics)
Laminar Transport Processes
Lagrangian vs. Eulerian Description
Material Derivative
Reynolds’ Transport Theorem
Conservation Laws
Conservation of Mass
Conservation of Momentum
Constitutive Relation
Navier-Stokes Equation

3. Importance of Viscous Phenomena
Objectives
Fluid dynamicist is generally engaged in the analysis or design of the practical devices such as an ship, airplane, a piping system and heat exchanger etc.
Primary consideration is the operation cost as well as performace.
Determine fluid forces acting on a body surface in a moving stream
ex) Drag of ship/airplane  Determine engine power (Heat/mass transfer)
Pressure drop in pipeline  Determine pumping power
Inviscid fluid assumption is too restrictive.
How can we do it ?
If we know velocity, pressure, density, temperature, concentration … as a function of space and time.
On what principle ?
Conservation law : mass, momentum and energy
Fluid mechanics is nothing but a kind of Newtonian Mechanics. (No Relativistic theory, No quantum mechanics !!!)

4. Conditions of at a Fluid-Solid Boundary (I)
The mathematical basis of modern fluid dynamics mostly before the true molecular nature of matter was well understood.
Thus, the fluid was considered a homogeneous, continuous medium. (Except for the highly rarefied air at outer edge of atmosphere)
For the boundary layer, the most important concern is the conditions at the interface between the fluid and a solid surface.
In terms of the scale of an air molecule, it is almost impossible to handle our purpose.
Fluid loses the all of its momentum relative to the surface and the fluid becomes to the same temperature as the surface.

5. Conditions of at a Fluid-Solid Boundary (II)
Let’s consider the velocity and temperature profiles yielded by above assumptions.
The velocity at the surface is imposed as the no-slip condition leading to a surface is a sink for fluid momentum.
Viscous layer (boundary layer) thickness δ is identified by the distance from the surface to 99% of Ue (edge or free-stream velocity)

6. Concept of Continuum Mechanics
What’s “continuum” ?
Fluid consists of continuous distribution of matter.
No molecules and vacuum between the molecules are considered.
At every point, the field variables (velocity, density, …) can be defined as a “volume average” with an infinitesimal small volume of fluid.
It’s a hypothesis.
Is “continuum hypothesis” really valid ?
For most of engineering problems with (mean free path) << (major dimension)
mean free path ~ spacing between each molecule ~10-8 m
The fluid particles exist nearly everywhere !
Exception : Rarified gas flow (too low pressure & density) for re-entry vehicles.
Turbulence ? What do you think about ?

7. Laminar Transport Processes
The simplest class of viscous flows is the laminar flow where the streamlines form an orderly and roughly parallel pattern.
Fluids move along the smooth paths
Viscosity damps any tendency to swirl or mix

8. How to describe the fluid motion?
Lagrangian description
Describe a defined fluid particle (position, velocity, acceleration, pressure …) as if we “fly along with” the particle as a function of time.
Example : track the location of a migrating bird.
Definitely useful if there is only a small number of particles (i.e. Dynamics of a multi-body motion, planets…).
In fluids mechanics : Vortex method.
Eulerian description
Describe the flow field (velocity, acceleration, pressure …) as if we “stay in fixed position” and wait for the passage of the fluid particle.
Example : count the birds passing a particular location.
Useful if there is so many number of particles (i.e. flow of sand particles…).
In fluids mechanics : Usual way of description, most CFD methods using grid.
Why the governing equation so complicated ?
In principle, it’s simply F=ma (Newton’s 2nd law), but using Eulerian description.

9. Material derivative (I)
Lagrangian rate of change
α : any field variable (velocity, density, pressure …)
Fluid particle moves between t and t+δt
Change in α during the time interval δt is
Rate of change : when δt  0 α
At time t
α+δα
At time t+δt

10. Material derivative (II)
Tensor : General notation for physical quantity
Scalar : Rank 0 tensor
Vector : Rank 1 tensor
Matrix : Rank 2 tensor
Tensor notation : useful for description of physical law with
Useful for description of physical law when scalar, vector and matrix are mixed.
Einstein’s summation convention : using repeated index, so simple !
Material derivative
In vector notation :
In tensor notation :

11. Reynolds’ transport theorem (I)
Control volume (CV) : a virtual volume in which we describe the field variables.
A moving system flows through the fixed control volume.
The moving system transports properties across the control volume surfaces.
We need a method to keep track of the properties that are being transported into and out of the control volume.
Lagrangean description  Eulerian description

12. Reynolds’ transport theorem (II)
Let’s consider the motion of fluid particle
α : any property (mass, momentum …)
Rate of change of volume integral of α during the time interval δt is
Change due to the change of V
Change within V
For superimposed control volumes at time t+δt, we need consider only volume change
using Gauss’ theorem

13. Reynolds’ transport theorem (III)
In summary
Rate of increase of the property in the system
Rate of increase of the property in the control volume
Rate of efflux of the property across the control volume boundary = +

14. Conservation Law : Mass
Using Reynolds’ transport theorem
If we let α = ρ (density of fluids)  Mass conservation
Rate of change of mass inside an arbitrary CV  Continuity equation
Simplification : Incompressible Flow
ρ = const.

15. Conservation Law : Momentum (I)
Using Reynolds’ transport theorem
If we let α = ρV (momentum of fluids)  Newton’s second law
Rate of change of momentum inside an arbitrary CV  Momentum equation
Force exerted on CV = Surface force + Body Force
Surface force in x1-direction
Generally, in xj-direction, including body force
Force exerted in
control volume

16. Conservation Law : Momentum (II)
Simpler case : 2-D elements

17. Conservation Law : Momentum (III)
Momentum equation becomes
Therefore, or
In each component : three equations
Solid mechanics : LHS (Acceleration) is zero.
There is no difference between solid & fluid.
=0 from continuity

18. Constitutive relation : Stress vs. Strain (I)
Let’s return to the definition of fluid.
When the external (shear) forces are exerted,
Solid : reaches equilibrium strain
Fluid : deforms continuously under the stress, can not sustain a shear stress when at rest.
Therefore, in fluid, Stress = function (rate of strain)
Cf : for solid, Stress= function (strain)
To solve the momentum equation, we will relate ‘stress tensor’ with ‘rate of strain tensor’.
And further (Stress tensor)  (Rate of Strain Tensor)  (Velocity Field).
The momentum equation will be described only with velocity.

19. Constitutive relation : Stress vs. Strain (II)
Rotation Rate and Shear Rate.
2-D Fluid element moves and deforms during δt
Angle deformation rate δα/ δt and δβ/ δt
Rate of rotation of fluid element
Rate of shear of fluid element
In general, for 3-D elements,
Rate of strain tensor is obtained from the velocity gradient
Symmetric part : Shear Rate
Anti-Symmetric part : Rotation Rate

20. Constitutive relation : Stress vs. Strain (III)
Stress = Normal stress + Shear stress
When fluid is at rest, the stress should be equal to the hydrostatic pressure (also to thermodynamic pressure).
Rate of rotation is not related with the shear stress tensor.
Assumption : Newtonian fluids
Stress tensor is linearly proportional to the rate of strain tensor.
Proportionality constant : Viscosity tensor (Rank 4 tensor with 81 elements)
Simplification : isotropy and (hydrostatic p) = (thermodynamic p)
Finally, we’ve got only one coefficient, viscosity μ.
Recall, in solid mechanics, we will get E (Modulus of elasticity) and ν (Poisson’s ratio).

21. Constitutive relation : Stress vs. Strain (IV)
Physical meaning of viscosity
In the 1-D case given below (plane Couette flow), there is only one stain rate in the flow.
In this flow field, shear stress τ12 is constant throughout the fluid layer. (Why?)
Therefore, ε12 should be constant.
For Newtonian fluid,

22. Constitutive relation : Stress vs. Strain (V)
Non-Newtonian fluids
Stress tensor is not linearly proportional to the rate of strain tensor.

23. Navier-Stokes Equation
Momentum equation + Constitutive Relation
For incompressible flow, or +

24. Simplification of Navier-Stokes Equation
Euler equation : neglect viscosity
How to derive Bernoulli's equation ?