Hüdro- ja aeromehaanika (1)

Tallinna Tehnikaülikool
Hüdro- ja aeromehaanika
EMH5020
Kodutöö
Üliõpilane:
Kood: XXX
Rühm: MATB-64
Juhendaja : Feliks Kaplanski
Kuupäev: 27.04.2012
Tallinn 2012
1. What means vorticity ? Derive vorticity transport equation in the plane and in the axisymmetric cases ?
1.1 Vorticity is equal to the curl of the flow velocity . Vorticity is the tendency for elements of the fluid to "spin."
 
Mathematically, vorticity is a vector field and is defined as the curl of the velocity field.
1.2 Vorticity transport equation on the plane
From the Navier - Stokes equation
We derive equation for two dimentional case on x-y plane (in the absence of gravity).
x component :
(1.21)
y component:
(1.22)
Now differentiating 1.21in respect to y and 1.22 in respect to x, so we get:
(1.23)
(1.24)
Subtracting (1.24) from (1.23)
(1.25)
Grouping this equation from 1.25:
(1.26)
Commutative property of partial differentiation
Continuity equation for incompressible flow:
Substituting to 1.26
(1.27)
On the x-y plane the vector field for velocity is defined:
When we calculate this curl, we get:
Subtituting to (1.27) we get the vorticity transport equation on the plane
1.3 Vorticity transport equation for axisymmetric case
Navier-Stokes equation:
We derive the equation for the axisymmetic case (in the absence of gravity):
z component:
(1.31)
r component:
(1.32)
Differentiating 1.31 in respect to r and 1.32 in respect to z, so we get:
(1.33)
(1.34)
Subtracting (1.33) from (1.34)
By grouping (1.35) we get
(1.36)
Vorticity for the axisymmetric case
Continuity equation for incompressible flow
By substituting the vorticity and continuity equation into (1.36) we finally get the vorticity transport equation for the axisymmetric case
9. Find velocity components for „ Kelvin cat eyes “:
We know that
and
, we need take derivation from given equation:
2. What advantages gives equations for vorticity transport and streamfuction in comparison with standard equations for velocity field? In what cases these advantages are often used?
Vorticity transport depends on the three Partial differential equations (PDEs) for u, v and p in the “primitive variable” form. Stream function depends on only two Partial differential equations (PDEs) for the scalars ω and ψ . We win on variables and this is the main advantage .
3. How will change vorticity transport equation if Reynold number will increase ?
Then Re number increase, it means, that we have turbulent flow. In this conditions Navier-Stokes equation take form of Euler equation. Navier-Stokes equation consist of two parts. One part , that consist volume becomes greater and equation transform into linear . Another part, what consist viscosity, becomes smaller and transform into non-linear term equation.
4. What means asymptotic analysing of the problem?
Asymptotic analysing is a method of describing limiting behaviour(boundary conditions). For example in physical system it describes behaviour in very large systems. We use this method then we operate with very high viscosity, much number and so on.

5. What gives introducing of Reynolds, Mach and other numbers in Fluid Dynamics?
This numbers help us to divide flow into different cases.
For example high Re number tell us, what we deal with turbulent flow. If Re number is low, we have laminar flow.
Much number is commonly used to represent the speed of an object when it is traveling close to or above the speed of sound .
Froude number is used to determine the resistance of an object moving through water, and permits the comparison of objects of different sizes. For Fr the flow is called a subcritical flow, further for Fr > 1 the flow is characterised as supercritical flow. When Fr ≈ 1 the flow is denoted as critical flow.
6. How can be investigated fluid flow with heating?

If we have changing in the temperature, we should use special type of equation (add equation for density ). In this case we have dependence of the density in coordinates. When changes are very small, we can neglect them .
7. What means Boussinesq’s approximation?
In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow (also known as natural convection). It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Sound waves are impossible/neglected when the Boussinesq approximation is used since sound waves move thru density variations.
8. Three laws of thermodynamics ?
First Law of Thermodynamics
The first law of thermodynamics is often called the Law of Conservation of Energy. This law suggests that energy can be transferred from one system to another in many forms. However , it can not be created nor destroyed. Thus, the total amount of energy available in the Universe is constant . Einstein 's famous equation (written below ) describes the relationship between energy and matter :
In the equation above, energy (E) is equal to matter (M) times the square of a constant (C). Einstein suggested that energy and matter are interchangeable. His equation also suggests that the quantity of energy and matter in the Universe is fixed .
Second Law of Thermodynamics
Heat can never pass spontaneously from a colder to a hotter body . As a result of this fact , natural processes that involve energy transfer must have one direction, and all natural processes are irreversible. This law also predicts that the entropy of an isolated system always increases with time. Entropy is the measure of the disorder or randomness of energy and matter in a system. Because of the second law of thermodynamics both energy and matter in the Universe are becoming less useful as time goes on. Perfect order in the Universe occurred the instance after the Big Bang when energy and matter and all of the forces of the Universe were unified.
Third Law of Thermodynamics
The third law of thermodynamics states that if all the thermal motion of molecules (kinetic energy) could be removed, a state called absolute zero would occur .
Absolute Zero = 0 Kelvins = -273.15° Celsius
The Universe will attain absolute zero when all energy and matter is randomly distributed across space . The current temperature of empty space in the Universe is about 2.7 Kelvins
10. Verify that the two-dimensional flow given in Cartesian coordinates by
satisfies , and then find the stream function such that and .
Constant value assigned to (x,0) is unimportant, because matter only gradients of
So
Equation for the streamfunction is:
9. If you put ice on one end of a metal rod, why does the other end get cold too?
Because of the heat exchange process , the metal rod is warmer than  ice and  therefore  the heat transferred to the ice, because of what an iron rod  begins to cool.
10. If you mix two colours of paint together, why can’t you unmix them?
We can’t unmix the paint as they have equal footing. When mixing the two colours is a diffusion  - the process of mixing (interpenetration) of the two substances, leading to self-alignment of their concentrations across the occupied volume. Since the diffusion process can proceed only in one direction is therefore impossible to obtain  again  two  original  paint.
11. How much does the humane body radiate? Estimate surface area A = 1.5 , e = 0.70
If the body temperature = 37 ºC = 37 +273 = 310 K,
surface area A=1.5,
emissivity e = 0.70
= Stefan Boltzmann constant = 5.67 x 10-8 W m-2 K-4
W
12. How much energy do we expend when we drink a glass of cold water and does that energy expenditure lead to weight loss if we drink more and more cold water?
A dieter's Calorie is the amount of energy to heat 1 kilogram of water 1 degree C.
If the water was 10 degrees C colder than body temperature and the person drank 0.5 liter of water, that person might expend about 5 Calories by drinking that glass of cold water.
If a person used water to reduce weight, I am sure that some of the water would be absorbed by the body to stabilize the excessive intake, perhaps resulting in losing fat but actually gaining weight.
13. Show that the Navier-Stokes equations for a two-dimensional, incompressible viscous flow can be write in terms of the streamfunction .
Viscosity and the Navier-Stokes equations a two-dimensional vorticity field . The idea is to prevent the vorticity from diffusing by placing it in a steady irrotational flow field of the form . Thus the full velocity field has the form
Now the z-component of the vortiity equation is, with
First note that if
= 0, so that there is no diffusion of , we my solve the equation to obtain
where
is the initial value of. This solution exhibits the exponential growth of vorticity coming from the stretching of vortex tubes in the straining flow .
If now we restore the viscosity, we look for a steady solution of
representing a vortex in for which diffusion is balanced by the advection of vorticity toward the z- axis .
We have
Integrating and enforcing the condition that and vanish when r = ,
we have
Thus
so that
where we have redefined the constant to exhibit the total circulation of the vortex. Note that as
decreases the size of the vortex tubes shrinks. With
fixed this would mean that the vorticity of the tube is increased.