Fluid Dynamics Flashcards
Notation for pressure and density
pressure; p = p(x, y, z, t)
density; ρ = ρ(x, y, z, t)
SCALAR FUNCTIONS
Steady flow
The flow is independent of time s.t.
u = u(x)
(where u and x are vectors)
Opposite of steady flow?
Unsteady flow
The flow is dependent of time s.t.
u = u(x, t)
(where u and x are vectors)
Define a stagnation point
A stagnation point is a location, x = x, say, at which u(x) = 0
(where u, x and x* are vectors)
What force does pressure (p) give rise to? And in what direction?
Pressure p gives rise to a force -∇p directed from regions of high pressure to regions of
low pressure.
By laws of vector calculus, what is the relationship between -∇p and the level surfaces?
The vector -∇p is perpendicular to the level surfaces, p = constant.
(NB p = constant is the definition of the level surfaces being referred to in this case)
What law of Newtonian dynamics is the Continuity Equation a result of?
The continuity equation results from one of the most fundamental laws of Newtonian dynamics,
that of CONSERVATION OF MASS.
Relationship between mass, volume and density?
M = V x ρ
Define the flux of a quantity through a surface.
The flux of a quantity through a surface is the rate of flow of the quantity through the surface, or
the amount of quantity that flows through the surface per unit time.
Another term for The Continuity Equation.
Mass Conservation equation
Another term for the Divergence Theorem
Gauss’ Theorem
Define the material derivative for scalar function Φ.
DΦ/Dt = ∂Φ/∂t + u · ∇Φ
Where u is a vector (velocity vector)
State the continuity equation in its material derivative form.
Dρ/Dt + ρ∇ · u = 0
Where u is the velocity VECTOR
What does Dρ/Dt mean in context?
Dρ/Dt is the rate of change of density following the fluid motion.
Define an incompressible fluid.
Incompressible fluids do not expand or contract. Therefore Dρ/Dt = 0 which implies ∇ · u = 0 (the incompressible form of the continuity equation)
Name for the suffix ‘i’
The free suffix.
NB ‘i’ is arbitrary and we could equally have used ‘j’ or another other suffix.
What is a repeated suffix referred to as?
As a dummy suffix.
It is summed over. A suffix that is free of summation is called a free suffix or a live suffix.
The trace of a matrix , A, in suffix notation
Tr(A) = A11 + A22 + A33 = Ajj
Is the Kronecker delta function symmetric?
Yes
δ_ij = δ_ji
Simplify δ_ij a_j
δ_ij a_j = a_i
What is the name for this; ε_ijk
The alternating tensor
Define the cross product using suffix notation.
(a x b)_i = ε_ijk a_j b_k
(NB a & b are vectors on the LHS, and this definition is for the ith component of the cross product of vectors a and b.)
What expression is helpful for remembering the relationship between the alternating tensors and the Kronecker delta function?
middle middle outside outside - middle outside outside middle
ε_ijk ε_ilm = δ_jl δ_km - δ_jm δ_kl
Newton’s second law
F = ma
F & a are vectors
F is force, m is mass and a is acceleration.
Relationship between acceleration and velocity
Du(x,t)/Dt = ∂u/∂t + u · ∇u = a(x,t)
In the Navier-Stokes equation, what does μ represent?
μ represents the coefficient of viscosity, which is a property of the fluid.
What does the viscous TERM in the Navier-Stokes equation represent in context?
The viscous term µ∇²u (u is the velocity vector) gives the resistance of fluid elements to being sheared.
Unless stated otherwise, what should be assumed about a fluid (to complete 5 equations for 5 unknowns)?
Take the density ρ to be a constant, i.e. ∂ρ/∂t = 0 and ∇ρ = 0.
The equation set then simplifies to that for constant density incompressible flow.
i.e. use ∇ · u = 0 as the continuity equation.
Give the two main categories of solutions to the Navier-Stokes equations.
Solutions for which the nonlinear term u · ∇u vanishes and those for which it doesn’t.
What types of flow are an example of when the non-linear term in the Navier-Stokes equation vansishes?
Couette flow
Poiseuille flow
Define streamwise
In the same direction as the velocity vector u.
No slip boundary conditions
Fluid at a boundary moves at the same velocity as the boundary
Regularity ‘boundary conditions’
We eliminate unphysical terms by an appropriate choice of constants
Free surface boundary conditions
The absence or neglect of forces on a surface (e.g. water in air) gives rise to a no stress boundary condition
µ ∂u/∂y = 0
Relationship between angular and linear velocity.
u = Ω x r
Where u is the linear velocity vector and Ω is the angular velocity vector and r is the position vector.
r = (x, y, z) in cartesian coords
r = (R, 0, z) in cylindrical coords
What is the trace of the symmetric matrix, e_ij (or if considering the trace e_ii), equal to?
The divergence of the flow.
∇ · u
Thus it it 0 for the incompressible flows.
If the trace is positive the fluid is expanding.
Define streamlines
dx/u = dy/v = dz/w
dx1/u1 = dx2/u2 = dx3/u3
Define vorticity
ω = ∇ x u
Relationship with velocity and the streamlines.
Velocity is tangent to the streamlines.
Can the vorticity be written in terms of the symmetric or anti-symmetric component of the Deformation matrix?
Anti-symmetric
In context, what do the symmetric and anti-symmetric parts of the Deformation matrix represent?
The symmetric part represents local straining motion
The anti-symmetric part represents local solid body motion
Explain the ‘modified pressure’ version of the Navier-Stokes equation
g = ∇(g · x) , where x and g are vectors
Modified pressure is represented by P
-∇p + ρg = -∇P
With P = p - ρg · x
Define a similarity solution
A similarity solution is a form of solution
that ‘looks the same’ at all times, or at all length scales
What are the two types of units for measuring physical quantities?
Fundamental units; Standard reference values, e.g. centimetres (length), grams (mass)
Derived units; Obtained by using the definitions of the physical quantities involved, e.g. speed (cm/s)
Define Dimension
The change in the numerical value of a physical quantity when one system of units is changed into another is determined by its dimension.
Difference between LMT and LFT systems?
F stands for Force
M stands for Mass
Define a dimensionless quantity?
Quantities whose numerical values
are identical in different systems of units are called dimensionless (they are said to have dimension 1)
Define independent dimensions
Quantities a1, a2, …ak have independent dimensions if none of the quantities have a dimension that can be represented as the product of powers of the dimensions of the others.
Note that none of a1, a2, …ak can be dimensionless (since 1 can be written as the product of the zeroth power of all of the other dimensions).
Name the dimensionless number in the dimensionless Navier-Stokes equation.
The Reynolds Number
Re = ρUL/µ
Categories of the governing parameters
Parameters with independent dimensions
And the parameters which don’t.
Dimensions of the kinematic viscosity
[ν] = L²T^{-1}
Define a boundary layer
Boundary layers are thin layers near/adjacent to the surface of a body in which strong viscous forces exist, due to large velocity gradients. The large velocity gradients are a result of the no-slip boundary condition which reduces the flow velocity to zero (on a stationary boundary).
Vorticity is generated in these layers.
How to obtain the vorticity equation.
Take the curl of the Navier-Stokes equation.
Viscous diffusion term
ν∇²ω
Where ω is the vorticity vector
Another term for stagnation point flow
Hiemenz flow
Describe the conditions for stagnation point flow
Consider a fluid flow whose velocity vector coincides with the y axis and impinges on a solid plane boundary y = 0
What condition on the Reynolds number allows for viscous effects to generally be neglected? (Exception?)
Re»_space; 1
Except in thin boundary layers adjacent to the body and in the downstream wake.
Blasius obtained an exact solution to the steady boundary layer equations for what conditions?
Uniform flow (stream)
u = (U,0)
where u is the velocity vector
Over a semi-infinite plane y=0, x>0
What are the off-diagonal elements of the stress tensor?
Shear stress
The diagonal elements are the normal stresses
Describe skin friction
Shear stress at the boundary
What does the displacement thickness measure?
Measure of the boundary layer thickness but in terms of the streamlines.
It is the amount by which a streamline in mainstream as x -> - infinity is displaced for x > 0
What two forces does the Reynolds number give the ratio of?
Inertia and Viscous forces
Inertia terms in N-S equation
Du/Dt = ∂u/∂t + u · ∇u
Another term for the stokes flow equations
Slow flow equations
Define 2D planar flow
2D planar flow is the flow assumed to flow only in a single plane with varying property at different points.
Curl of grad?
… x ∇ =
Zero
… x ∇ = 0
Finish the identity for Cartesian coordinates
∇ x ∇²F = ….
(F is a vector)
∇ x ∇²F = ∇²( ∇ x F )
NB F is a vector
State the biharmonic equation
∇^4ψ = 0 in the planar case E^4ψ = 0 in the spherical case
∇ · ω = ?
∇ · ω = 0
Stoke’s drag on a sphere
Total force
on the sphere
How to find (plane) polar coordinates from cylindrical polar coordinates?
Set z = 0
Write the gravity vector in terms of its potential
g = -∇π
π ~ gravitational potential, scalar
g is a vector
curl of the gradient…
is zero
use analogy, write intials of curl, divergence and gradient in alphabetical order;
c d g
arrow from c to g
arrow from d to c
divergence of the curl…
is zero
Describe what Dω/Dt is in context?
The rate of change of vorticity experience by a moving fluid parcel.
From the vorticity equation; the rate of change of vorticity as experienced by a moving fluid element is due to the
effects of velocity gradients and the effects of viscosity.
Define a pathline
The trajectory of a fluid parcel as it moves with the flow
Define a vortex line
a curve drawn in the fluid which is everywhere tangent to ω. Defined in an analagous way to streamlines
Define a vortex tube
A vortex tube is the surface formed by all vortex lines passing though a closed curve C in the fluid
Kelvin’s Circulation theorem
The circulation around any
closed material contour is constant for inviscid flow.
First Helmholtz Law:
Fluid elements having zero vorticity…
Continue to have zero vorticity
By Dω/Dt = ω · ∇u
Since if ω = 0 at t = 0 on a fluid element, then ω = 0 is the solution at later times.
Second Helmholtz Law; Vortex lines are…
Material lines.
ω are transported like material line elements since they are defined by identical equations; D(dr)/Dt = dr · ∇u
(where dr is the vector of any line element)
Third Helmholtz Law;
The strength of a vortex tube is…
Constant
If the dot product between 2 vectors is 0, what can be deduced?
The vectors are perpendicular
Another way to think of incompressibility in context
Volume preserving
Incompressible fluids conserve their volume
Strain flow
We consider ‘strain’ flows in which infinitesimal vectors are stretched and so we have vorticity intensification.
The name of the solution to the governing equation for vorticity (vorticity equation) in strain flows, where time is neglected but viscosity is included.
Burgers Vortex
Viscous, steady-state solution
What are waves and how are they generated?
Waves transport information between two points in space and time. Waves are generated due to the existence of a restoring force that tends to bring the system back to its undisturbed state, and some kind of inertia that causes the system to overshoot after the system has returned to its undisturbed state.
Define surface gravity waves
The wave that occurs at the free surface of a liquid, where gravity plays the role of the restoring force. These are called surface gravity waves.
Define wavenumber
The number of complete waves in a length 2π
Relationship between the wavenumber and the wavelength?
λ = 2π/k
Where λ is the wavelength and k is the wavenumber
Phase speed
c = ω / k
The rate at which the ‘phase’ of the wave (crests and
troughs) propagates.
Define period T of a wave
The period T of the wave is the time required for the wave to travel one wavelength
T = λ/c
(may not be the speed at which the envelope of a group
of waves propagates - the group speed)
Another term for plane waves
Monochromatic waves
Define a free surface of a fluid
The surface of a fluid that is unconstrained from above
Define inviscid
Having no or negligible viscosity
Define irrotational flow
No vorticity
ω = ∇ x u = 0
What does the velocity potential ensure?
vorticity is zero
What is the kinematic condition (in words)?
The surface moves with the
fluid
What is the dynamic condition (in words)?
The dynamic condition arises from a condition on the pressure at the free surface.
For inviscid flow in the presence of gravity the Navier-Stokes equation reduces to Euler’s equation.
Where does the difficulty in solving surface-wave problems arise from?
The difficulty in solving surface-wave problems is due to the boundary conditions rather than the differential equation.
In the study of surface gravity waves, where does ‘Laplace’s equation’ come from?
The incompressibility condition on the velocity potential
What is the dispersion relation?
Gives the speed of the propagation of the wave.
Connects the frequency and the wavenumber, ω = ω(k).
Dispersion means that waves of different wavelengths travel at different speeds.
Define group speed
The group velocity is the velocity at which
an isolated wavepacket travels as a whole, and it is the velocity at which energy is
transported.
Non-dispersive waves
The phase speed is independent of wavenumber, k. Waves of different wavelengths travel at the same speed
