Notes 4: Differential Form of the Conservation Equations

Transcription

1 Low Speed Aerodynamics Notes 4: Differential Form of the Conservation Equations

2 Deriving Conservation Equations From the Laws of Physics Physical Laws Fluids, being matter, must obey the laws of Physics. The ones we need are: 1. Conservation of Mass: Mass is neither created nor destroyed (nor changed to anything else) 2. Newton's Laws of Motion 1st Law: concept of equilibrium 2nd Law: force, acceleration, momentum 3rd Law: action & reaction Conservation of Energy -- 1st Law of Thermodynamics These must be valid for all flow problems that we consider. They might need modification if the flow is inside a star or a Black Hole, where there may be substantial conversion between matter and energy Also, for problems involving spacecraft (or atoms or subatomic particles) accelerating to speeds close to that of light. ategorized/2007/07/05/black_hole_3_3.jpg

3 Constitutive Relations In addition, we can get relations which are specific to the kind of fluid with which we are dealing. These equations are called the "Constitutive Relations". They include the equations of state. Equations. of State [relations between different properties] p Perfect Gas Law: Thermal eqn. of state Caloric eqn. of state Energy contained per unit mass of a substance = (specific heat)*(temperature) The independent variables are: velocity, density, pressure and temperature. To solve for these in a given problem, we have the 3 conservation equations of mass, momentum and energy, and the thermal state equation. In addition, we may be interested in other variables, like the speed of sound. Here the properties of the fluid are taken into account further using the caloric equation of state. t

4 Conservation of Mass If we have a fluid going g in and coming out of a given region of interest, (a "control volume"), we can say for sure that what goes in per unit time = what comes out per unit time + what accumulates inside per unit time. Mass going in per second= { Sum of masses going out per second + mass accumulated inside in 1 second. } In general, mass may be going in and/or out everywhere across an (imaginary?) surface enclosing the space in which you are interested. Also, the velocity of the inflow/outflow may be nonuniform, and in some arbitrary direction.

5 Mass flowing out of the surface of the control volume per unit time = where ds is a small element of the total t surface area S. S ds u n An "integral" is a neat way of saying, collect all the little bits and add them up." Law of conservation of mass for a control volume becomes S u n ds dv 0 t V Conservation" of anything can be expressed this way. as we will see. Here the r.h.s. is zero: mass can't get converted to anything.

6 Example: Closed circuit wind tunnel Settling Chamber velocity = 15 fps, uniform; steady. Chamber diameter: 20 feet. Test section is a 9-foot diameter duct. Our "control volume" has one face in the settling chamber, other face in the test section. Assume that no flow can escape out the sides of the control volume. "Steady" means that at any given point in the flow, the properties don't change as time changes. ( ) 0 t anything The integral form of the mass conservation equation becomes: ds u n 0 Subscript "1" refers to settling chamber face of control volume, subscript "2" refers to test section. Air cannot escape out of the sides of the "control volume, so the integral reduces to just -ρρ 1 A 1 U 1 + ρ 2 A 2 U 2 = 0 Negative sign: Look at direction of vector normal to each face, with respect to the direction of U. Thus U 2 = (ρ 1 A 1 U 1 ) / (ρ 2 A 2 ) Since the density change can be neglected, and each area is a circle, U 2 S = (20/9) 2 *15 = ft/s.

7 Conservation of Momentum Newton's 2nd law of Motion: "Rate of Change of Momentum = Force" or "Net force acting on a system = time rate of change of momentum of system" Note units: Momentum = mass x velocity = density x volume x velocity = density x area x distance x velocity Momentum per unit time = density x velocity x velocity x area What is being conserved? "momentum per unit volume". So the conservation equation has 2 terms on the left hand side: 1. "rate of change of the quantity, integrated over the whole control volume". This is the "unsteady term". 2. "net outflow per unit time of the quantity across the surfaces of the control volume, integrated over the whole control surface". This is the convection term. On the right hand side, we have surface and volume integrals of the quantities to which our "conserved quantity" may have changed. In this case, when momentum disappears, the rate of change of momentum produces forces of various kinds.

8 In general, we consider two kinds of forces: Forces a) "Body forces" : forces that act per unit mass, such as gravity or electromagnetic forces. b) "Surface forces": forces that act per unit area acting normal to the area and parallel to b) Surface forces : forces that act per unit area, acting normal to the area and parallel to the area.

9 Types of Forces to be included in the momentum equation a) Body forces Gravitational force per unit volume = g (Important for liquid flows, and buoyant gas flows). Electromagnetic force: it would be something like Important in ion propulsion, spark plugs, plasmas. Generally, we neglect body forces in aerodynamics. b) Surface forces B Pressure is the result of molecules, flying about at random, crossing [ or colliding with ] the surface, thus transferring momentum. So it acts normal to the surface. It must be related to # of molecules/volume [thus related to density], and speed of random motion of molecules [thus related to temperature]. Thus, pressure is related to the product of density and temperature. This relation is a simple proportionality (multiply by a constant) where R is the gas constant. This is called the "perfect gas law." So force due to pressure is n S p ds because pressure acts on the surface, opposite to n

10 Viscous Forces Fluids, like other forms of matter, resist movement. Part of the resistance is explained by pressure: this part is reversible. The rest is irreversible: when fluids move, there is some loss. This is attributed to "viscosity." There must be some relation between stress and strain. In solids, stress is a function of strain. In the simplest case, stress is directly proportional to strain Fluids don t resist shear strain if applied slowly. The resistance is to the rate of strain. Thus, ideally y( (this is called Newtonian Fluid ), in fluids, stress is proportional to Rate of Strain. The proportionality constant is called Absolute Viscosity.

11 xy u y v x yz v z w y zx u z w x Shear Stress vs. Rate-of-Strain Relations The direction is indicated by the second subscript. The units of stress are Force/Area, such as Newtons/m 2, psi, or psf. These can be written in compact form. Here u is used to represent any of the velocity components, and x is used to represent any of the spatial coordinates. i and j representing x,y,z in turn. u j u i ij xi x j The normal stress, on the other hand, is simply the pressure.

12 Normal Stress Here normal is a fashionable term for perpendicular to a surface, as opposed to tangential, not as opposed to abnormal.. The normal stress is simply due to the pressure. xx pi yy pj zz pk The unit vectors are included to show the direction of each of these stress components. Don t fluids resist the strain rate due to compression and expansion? They do, and a relation between normal stress and rate of normal strain can be written, just like the shear strain relations on the previous slide, and added on to the above expressions. However, in gases, the coefficient of resistance to normal rate of strain is very small, so that this bulk viscosity does not become significant unless the strain rate is extremely high. This occurs, for instance, across a shock wave, where velocity decreases by a large factor in the space needed for a few (under 10) collisions between molecules. In most other situations, it is a usual assumption that Bulk Viscosity is Zero in a gas so that the, p y g normal stress is attributed to just the pressure.

13 Energy Equation In addition to mass and momentum, we can use the fact that energy is neither created nor destroyed, but can change form The conserved quantity is energy per unit volume. The lhs has the terms describing time rate of change, and flux across the surfaces of the control volume. The rhs has the terms describing what changes the energy in the control volume: work and heat transfer. We use the first law of thermodynamics, which says that if you do work or release heat, your energy is exhausted. We will leave detailed study of this equation to the course on compressible flow and thermodynamics. The measure of total energy in a flow is the stagnation enthalpy, which is related to the stagnation value of temperature. So the energy equation reduces to: "Stagnation temperature goes up if work or heat are added to the flowing fluid". In steady flow, with no work being added, and no heat being added (adiabatic flow), the rhs is zero, so that the energy equation reduces to Stagnation Temperature is Constant : T 0 T u2 constant 2c p p The cp here is not pressure coefficient, but specific heat at constant pressure. c R p 1 For diatomic gases such as air, at usual temperatures that we encounter in low speed aerodynamics, the gamma has the value 1.4. Thus c p = 3.5R

14 Why we don t worry about the energy equation too much here In low speed aerodynamics, density is usually assumed constant (changes due to velocity changes are negligibly small). Then you need one less equation because density is no longer an unknown variable. Thus the energy equation is rarely needed, d except when one worries about what happens when work is added to the flow, as by a propeller or rotor. In such cases we will relate the stagnation pressure to the stagnation enthalpy In such cases, we will relate the stagnation pressure to the stagnation enthalpy, and use that instead as a measure of work added or removed.

15 Fluid Dynamics Summary 1 1. Assumed that fluid flow is a continuum phenomenon. 2. Fluids, being composed of matter, obey the laws of physics. To calculate the lift and drag forces, we would like to be able to calculate the pressure acting on the surface. This depends on the velocity, and also the state of the fluid, which requires knowledge of the temperature and density. Thus, we would like to be able to calculate the velocity vector, density, temperature and pressure everywhere, as functions of space (x,y,z) and time (t). For this we use 3 laws of physics: Mass is neither created nor destroyed... (1) Rate of change of momentum = Net force...(2) Energy can change form, but is neither created nor destroyed...(3) In addition, the state of the gas is specified by the State Equations. The Thermal Equation of State for a gas reduces to the Perfect Gas Law, which relates density, temperature and pressure. To solve very any flow problem, we can specify the boundary conditions and/or initial conditions, and solve all these equations simultaneously all over the flow field.

16 In the next sections, we will see how to reduce these conservation equations to forms which can be used to calculate l changes in velocity and pressure over space and time, and thus to calculate l flow properties at given points, or over entire vehicles.