Governing Equations of Fluid Flow Session delivered by: M. Sivapragasam 1
Session Objectives -- At the end of this session the delegate would have understood The principle of conservation laws Different principles involved in simplifying approximations The equations governing the fluid flow The role of non-dimensional o equations Various simplified equations 2
Session Topics 1. Conservation Laws 2. Simplifying Approximations 3. Equations of Fluid Flow 4. The Stress at a Point 5. TheRoleofNon-dimensional of Equations 6. Various Simplified Equations 7. Bernoulli Equation and Examples of Pressure Distribution 3
Conservation Laws To derive the equations of motion for fluid particles we rely on various conservation principles. These principles are entirely intuitive. They are a statement of the fact that the rate of change of mass, momentum, or energy in a certain volume is equal to the rate at which it enters the borders of the volume plus the rate at which it is created inside. The first two of these will be used extensively here. 4
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These integral expressions are combined with the divergence theorem and the fact that they hold over arbitrary volumes to obtain the differential form of the equations:.. 6
We can use the momentum theorem by itself to obtain useful results. In this example, we apply the momentum theorem to relate the force on a body to the properties p of the flow some distance from the body. This technique is useful in wind tunnel tests and is the basis of several fundamental theorems related to lift and induced drag of wings. We take the control volume shown below, bounded by the single surface, S which hwe divide id into 3 parts: the outer surface (S outer ), the inner surface (S inner ), and the pieces of the surface connecting the two (S*). We can write the integral form of the momentum equation for steady flow with no body forces as shown. 7
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Note that the contribution from the part of the surface connecting S inner and S outer to the integrals is zero because as the two pieces of S* are made close together, th the unit normals point in opposite directions while p and V are equal. 9
Simplifying Approximations The equations of motion for a general fluid are extremely complex and even if the problem could be formulated it would be impractical to solve. Thus, from the outset, certain simplifying approximations that are often very accurate, are made. These may include the following assumptions. 10
Continuity and Homogeneity of the Fluid We assume that the fluid is composed of particles which are so small and plentiful l that t the statistically-averaged ti ti properties of interest are the same at any scale. This works well for gases and fluids under most conditions. It does not work for studying the flow of sand. It does not work when the fluid is so rarefied that the mean free path is of the same order as the dimensions of interest in the problem. The mean free path varies with altitude as shown in the next plot. Variation of temperature, pressure and density with altitude is also shown. We further assume that t the medium can be treated as a single type of fluid -- no suspensions of oil and water. 11
Variations of temperature, pressure and density with altitude. From: Kuethe &Ch Chow. 12
Variations of mean free path λ with altitude. From: Kuethe & Chow) 13
Inviscid idflow The effect of viscosity may sometimes be neglected or modelled indirectly. For many aerodynamic flows of interest, the region of high shear and vorticity is confined to a thin layer of fluid. Outside this layer, the fluid behaves as if it were inviscid. Thus the simpler equations of an inviscid fluid are often solved outside of the shear layers. There are some fluids which seem to be almost completely inviscid. Tests in superfluid helium have given results similar to inviscid calculations. 14
Incompressible (constant density) Flow PEMP When the fluid density does not change with changes in pressure, the fluid is incompressible. Water density changes very little with changes in pressure and is generally treated as an incompressible fluid. Air is compressible, but if pressure changes are small in comparison with some nominal value, the corresponding changes in density are small also and incompressible equations work quite well in describing the flow. The degree to which the fluid density changes with pressure is related to the speed of sound in the fluid. Thus, assuming that the flow is incompressible is equivalent to assuming that the speed of sound is infinite. When the local Mach number is less than 0.2 to 0.5 compressibility effects can often be ignored. The reason for this is discussed further in the chapter on compressibility, but one can see qualitatively that in order to make an appreciable change to the nominal 1 bar air pressure at sea level, substantial speeds are required. 15
Irrotational Flow Circulation is defined as: PEMP It is a measure of the rotation of an area of fluid. As the integration contour is shrunk down to a point, the ratio of circulation to the area enclosed by the curve is called the vorticity. Fluid that t starts t out without t rotational ti motion will not develop it unless there has been some shear stress acting on it. Some important exceptions to the idea that without viscosity irrotational flow remains irrotational: Vorticity can be created in a gravitational field when density gradients exist or in a rotating system (such as the earth) due to Coriolis forces. These are important sources of vorticity in meteorology. 16
And if the shear is confined to a small region, the vorticity it will be also. Thus, for many cases, especially in inviscid flow, much of the flow field may be treated as irrotational: curl V = 0 When this is the case, the vector field, V, may be written as the gradient of a scalar field, φ: V = grad φ where φ is the called the potential. This simplifies many of the equations discussed in subsequent sections. The velocity components are then: u = d φ / dx and v = d φ/ dy 17
Steady Flow When the variables describing the fluid properties p at a given point do not change in time, the flow may be treated as steady and the time derivatives in the equations of motion are zero. This condition depends on the chosen coordinate system. If the system is at rest with respect to a body in uniform motion through a fluid the equations in that system are steady, but expressed in a system fixed with respect to the undisturbed fluid, the flow is unsteady. It is often convenient to transform the coordinate system to one in which the flow is steady. This is, of course, not always possible. We will assume that the flow is steady in most of the discussions in this course but unsteady effects are often important in the study of bird flight, propellers, aircraft gust response, dynamics, and aeroelasticity as well as in the study of turbulence. We will always apply the first of these assumptions and will sometimes adopt one or more of the latter in the following discussions. 18
Equations of Fluid Flow The conservation laws may be used to derive the equations of fluid flow. These are supplemented with constitutive relations such as the perfect gas law: p = ρ R T or the isentropic relation between pressure and density: p /p = γ 2 1 (ρ 2 / ρ 1 ) We start with the Navier-Stokes equations and show how non- dimensionalising these equations leads to the non-dimensional parameters like Re and M. We then identify some of the most commonly-solved equations along with the corresponding assumptions. 19
The Navier-Stokes Equations The Navier-Stokes equations describe the flow of a continuous, Newtonian fluid. They may be derived from the principal of conservation of momentum applied for a field for an identified fluid particle. Hence we use the substantial derivative D( ) / Dt given by D( ) Dt ( ) ( ) ( ) ( ) ( ) r = + u + v + w = + V. ( t x y z t (For more details see Kuethe and Chow Appendix B, or, Moran Ch. 6, or, Anderson Ch. 15). ) 20
Here X, Y, Z are the body forces per unit mass in each direction and Tau is the stress tensor. X,Y, and Z are often associated with gravitational forces and are often neglected. ec ed. 21
The equations contain unknown stress terms τ xy and hence cannot be solved (not closed). They can be closed when the stress tensor is expressed in terms of viscosity, pressure and velocity (derivatives). Then the N-S equations simplify. 22
The stress at a point PEMP τ xy is the stress acting on the x-plane in the y-direction. Pressure is a normal compressive stress. μ (shear viscosity relating stresses to rate of strain) and λ (bulk viscosity relating stresses to div V) are the two viscosity coefficients. ). If pressure is a function only of density and not of the rate of change of density, then: λ = - (2/3)μ. Normal stresses: 23
See that now pressure in a moving fluid is the average of three normal stresses. Shear stresses depend only on μ : 24
S T i N i fl id Stress Tensor in a Newtonian fluid + = = + = u v u V μ σ σ μ λ σ ; 2 ) ( r + = = + = + = = + = w u v V y x x V yx xy xx μ σ σ μ λ σ μ σ σ μ λ σ ; 2 ). ( ; 2 ). ( r + = = + = + + w v w V x z y V zy yz zz zx xz yy μ σ σ μ λ σ μ σ σ μ λ σ ; 2 ). ( ; 2 ). ( r y z z zy yz zz μ μ ; ) ( 25
The Navier-Stokes equations for an incompressible fluid simplify to And the continuity equation for the incompressible flow (unsteady also) is u v w + + = 0 x y z 26
The Navier-Stokes equations in cylindrical coordinates (r, θ, z) are: 27
The Role of Non-Dimensional Equations PEMP The order of terms in the equations can be conveniently estimated by non-dimensionalising the equations. The procedure is demonstrated with simple incompressible equations. In the energy equation Φ is the viscous dissipation function. We start with the dimensional equations: 28
The equations are non-dimensionalised by non- dimensionalising i i independent d and dependent d variables through characteristic variables like V, L, ρ 1, a 1, T 1 etc. Here subscript 1 indicates some reference undisturbed flow conditions like at infinity. The new variables indicated by a prime are nondimensional. PEMP 29
Non-Dimensional Equations PEMP See how Reynolds number, Mach number and Prandtl number originate in the equations. It is clear from the momentum equation, for example, that when Re increases, the relative magnitude of the viscous term decreases. It is possible to infer this now since all the terms are scaled and it is possible to estimate. For details see Kuethe & Chow (Appendix B). Note ( Φ / μ 1 ) is non-d in this notation. 30
Solutions of the full Navier-Stokes equations show the onset of turbulence, the interaction of shear layers, and almost all of the interesting aerodynamic phenomena (with the exception of interacting or rarefied gas flows). Unfortunately, the equations are very difficult to solve. As the Reynolds number is increased, the scale of the interesting dynamics gets smaller so that most solutions of the full N-S equations are done at Reynolds numbers of 1 to 10,000 and for simple geometries. One of the most recent solutions of a flat plate boundary layer pushed the calculations to a Reynolds number of 1410 based on boundary layer thickness. These calculations took hundreds of hours on the Cray computer at NASA Ames. 31
Even at very small Reynolds numbers, the geometries which can be analyzed using the full N-S equations are quite simple and it currently does not make sense to consider solving these equations for realistic aircraft configurations. One reason that this is the case is that many of the approximate equations work quite well in such cases and are much more easily solved. 32
When the time averaged Navier-Stokes equations are not a sufficient description of the problem, one may resort to "large eddy simulations". This is a numerical solution of the time-dependent Navier-Stokes equations, with only the smaller scales of turbulence modeled in an averaged way. Larger scale turbulent motion can be included in this way. While this is faster than solving the full equations, it is still very slow. The figure below shows results from a large eddy simulation of the flow over a 2D circular cylinder. Each simulation required approximately 300 CPU hours and about 10 megawords of core memory on the Cray C-90. Figure from NASA / Parviz Moin. 33
Flow around a Cylinder simulated using Large Eddy Simulation (LES). From: NASA 34
Reynolds Averaged dnavier-stokes Equations One of the most popular pp simplifications made to the Navier- Stokes Equations is "Reynolds Averaging". This simplification to the full Navier-Stokes equations involves taking time averages of the velocity terms in the equations. Writing: u = <u> + u ', v = <v> + v', etc. (where < > represents a time average) with the fluctuations having zero mean value: <u'> = 0 we have: <u 2 > = <u> 2 + <u' 2 >, <uv> = <u><v> + <u'v'> 35
Reynolds Averaged dnavier-stokes Equations (Contd) This allows us to write the time-averaged NS equations as: and similarly for the y and z components. 36
Reynolds Averaged Navier-Stokes Equations (Contd) This looks just like the more general Navier Stokes equations for incompressible flow which hold for steady, laminar flow except that there are additional terms that act as additional stresses on the right hand side. These terms represent the effect of turbulence on the mean flow. They are called "Reynolds stresses" and are sometimes said to be caused by "eddy viscosity". These terms are generally much larger than the normal viscous terms. 37
The business of predicting these stresses and relating them to the computed mean flow properties is called turbulence modeling. This is usually accomplished empirically or by using the results of detailed timedependent simulations. Reynolds averaged NS solvers are appropriate for the analysis of viscous, compressible flows and have been applied to rather general configurations, i but one must be careful that the assumptions of the turbulence model are compatible with the characteristics of the flow of interest. 38
From NAS Technical Summaries, High-Lift Configuration CFD, Karlin Roth, NASA Ames Research Center 39
Euler Equations PEMP The momentum equation is sometimes called Euler's equation. (There are lots of equations called Euler equations!) But when people talk of solving the Euler equations these days, they are referring to the inviscid equations of motion given by: With some work*, the equation in the x direction becomes: or in vector notation: 40
These are combined with the equations of energy and continuity. The equations are often solved by finite differences whereby the values of each velocity component, the density, and the internal energy are computed at each point in the flow. From these quantities constitutive relations such as the perfect gas law or the isentropic pressure relation are used to find pressure. Since Euler equations permit rotational lflow and enthalpy losses (through shock waves), they are very useful in solving transonic flow problems, propeller or rotor aerodynamics, and flows with vortical structures in the field. 41
Mach number (Velocity magnitude?) contours by Euler equations. NOTE: Surface velocity is not zero. 42
Full Potential Equation The full potential equation is derived from the assumption of irrotational flow and the equations of continuity and momentum. The pressure and density terms in the Euler equations can be combined when use is made of the perfect gas law and the isentropic relation between pressure and density. Ashley and Landahl show how we may derive the following vector form of the unsteady full potential equation: 43
Full Potential Equation (Contd) This may be simplified for the case of steady flow in 2-D to: About the notation: When flow is irrotational curl V = 0 and by definition of curl and gradient: curl (grad φ) = 0 where φ is a scalar field. 44
Full Potential Equation (Contd) Thus we can define a nonphysical scalar potential, φ, that describes the velocity field. φ is related to the velocities by the relation: V = grad φ The equations can thus be written in terms of the unknown scalar rather than the 3 components of the velocity. This simplifies their solution. In the above expressions: a is the local speed of sound, x is the streamwise coordinate, and V is the vector velocity. Subscripts denote partial derivatives with respect to the subscripted variables (e.g. U x = du/dx) 45
Mach contours (yellow = high; dark green = low) from TRANAIR analysis of a 747-200 at Mach 0.84. From NAS Technical Summaries 1994, Multipoint Aerodynamic Design, Forrester T. Johnson, Boeing Commercial lairplane Group. 46
Transonic Small Disturbance Equation When the full potential equation is simplified by assuming that perturbation velocities are small and we relate the local speed of sound dto the freestream value by making use of fthe isentropic i relations we obtain the small disturbance equation When we let the freestream Mach number go to one and ignore the last term, the equation becomes the classic transonic small disturbance equation: A great deal has been written about this nonlinear equation and its variants. (See Nixon.) It is used less frequently these days since finite difference methods can be used to solve the full potential equation directly.. 47
Prandtl-Glauert Equation The Prandtl-Glauert equation is a linearized form of the full potential equation. PEMP Full potential: If the velocity perturbations are much smaller than the freestream velocity, this expression becomes: or in the unsteady case: The 3-D version is easily constructed with the addition of z derivatives corresponding to the y derivatives shown here. 48
Note that this linearized form of the equation does not hold near the nose of an airfoil where the velocity perturbation is of the same order as the freestream, unless the freestream Mach number is itself small. Also note that t this expression holds for subsonic and supersonic flow (but not transonic flow). It forms the basis for many aerodynamic analysis methods. Analysis of P51 Mustang from Analytical Methods, Inc. using VSAERO, a code that solves the Prandtl-Glauert equations. 49
Acoustic Equation The acoustic equation may be obtained from the full potential equation by assuming that there is no freestream velocity, and that all perturbation velocities are small. PEMP Or, by changing from a coordinate system fixed to the body to one fixed with respect to the undisturbed fluid, the Prandtl- Glauert equation may be transformed to the acoustic equation. This equation is often used in the study of sound propagation and sometimes for rotor aerodynamics; thus the name. 50
Laplace's Equation Laplace's equation is the Prandtl-Glauert equation in the limit as the freestream Mach number goes to zero. It was actually first derived by Euler. The derivation is very simple, requiring only the equation of continuity, and the assumptions of irrotational and constant density flow. The continuity equation becomes then: Since the flow is irrotational: Substitution into the continuity equation yields: 51
Laplace's Equation (Contd) It is interesting to note that Laplace's equation does not require the assumption of small perturbations, while the Prandtl- Glauert equation does. In fact, near the stagnation point of an airfoil where velocities become small, the full potential equation reduces to Laplace's equation, not the Prandtl-Glauert equation. Note also that all of the time dependent terms in the full potential equation are multiplied by 1/a 2 so that this form of the equation holds for unsteady phenomena as well. 52
Bernoulli Equations Some of the equations we have discussed are posed in terms of state variables that do not include pressures. In these cases (e.g. the potential flow equations) the differential equations and boundary conditions allow one to compute the local velocities, but not the pressures. Once the velocities i are known, however, the momentum equation can be used to find the local pressure. Such equations are known as Bernoulli equations and they come in various forms, depending on the assumptions that can be made about the flow. 53
Bernoulli Equations (Contd) The conservation of momentum principle p is the source of the relation between pressure and velocity. It can be used very simply to derive the Bernoulli equation. To illustrate the basic physics behind the Bernoulli equations, we can derive a simple form: that for steady, incompressible flow. In this case we show that along a streamline: 54
When the flow is not steady, the Euler equations can be integrated to obtain a more general form of this result: Kelvin's equation, the Bernoulli equation for irrotational flow. where f is a body force per unit mass (such as gravity) and F is an arbitrary function of time. 55
If we do not assume that the flow is irrotational, we cannot introduce the potential and the expression is not so nicely integrable. If, however, we assume that the flow is steady with no "body forces", but not necessarily irrotational we can write the following expression that holds along a streamline: While the above equations hold for steady flows along a streamline, for irrotational flows they hold throughout the fluid. 56
We can derive a more useful form of the Bernoulli equation by starting with the expression for steady flow without body forces shown just above. If the flow is assumed to be isentropic flow (no entropy change or heat addition): p = constant * ρ γ Substitution yields the compressible Bernoulli equation: This actually works for adiabatic flows as well as isentropic flows. 57
In summary, we often deal with one of two simple forms of the Bernoulli equation shown below. 58
The Pressures In both the incompressible and compressible forms of Bernoulli's equation shown above there are 3 terms. The quantity p T is the total or stagnation pressure. It is the pressure that would be measured at points in the flow where V = 0. The other p in the above expressions is the static pressure. Note that in incompressible flow, the speed is directly related to the difference in total and static pressure. This can be measured directly with a pitot-static probe shown below. 59
The Pitot-Static Probe PEMP The Ventui Tube 60
The dynamic pressure is defined as: The static pressure coefficient is defined as: where p is the freestream static pressure. In incompressible flow, the expression for C p is especially simple: If the local velocity is expressed as a small perturbation in the freestream: Then the incompressible C p relation can be written: 61
The expression for C p in compressible isentropic flow (sometimes called the isentropic pressure rule) is derived from the compressible Bernoulli equation along with the expression for the speed of sound in a perfect gas. In terms of the local Mach number the expression is: 62
We can tell if the flow is supersonic, just by looking at the value of C p. The critical value of C p, denoted C p * is found by setting M = 1 in the above expression (for gamma = 1.4): Also, we see that there is a minimum value of C p, corresponding to a complete vacuum. Setting the local Mach number to infinity yields: C p cannot be any more negative than this. Experiments show that airfoils can get to about 70% of vacuum C p. This can limit the maximum lift of supersonic wings. 63
Simple Bernoulli Derivation The momentum equation for this flow can be written in terms of the flow through h a small control volume. The change in momentum per unit time is: ρ S V (V+dV) - ρ S V 2 = ρ S V dv This change in momentum arises from the pressures acting along the faces of the control volume: pressure force (ends) = ps - (p+dp)(s+ds) = -p ds - S dp pressure force (sides) = (p + dp/2) ds = p ds (to first order) pressure (total) = -S dp 64
Simple Bernoulli Derivation (Contd) Equating the force due to pressure with the force required to produce the momentum change yields: -S dp = ρ S V dv or dp = -ρ V dv This is a simple form of the Euler equation. In the case that ρ = constant, the above equation may be integrated to produce the incompressible form of the Bernoulli equation: p 2 + ρ/2 V 22 = p 1 + ρ/2 V 2 1 or: p + ρ/2 V 2 = p t 65
Summary The following topics were dealt in this session Conservation laws Different simplifying approximations and the resulting equations The role of non-dimensional equations Bernoulli s equation and eh pressure distribution 66
Thank you 67