Hutton - Fundamentals of Finite Element Analysis (523155), страница 55
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In a very practical sense, viscosity is a measure of the “thickness” of a fluid. Consider the differences encountered in stirring a containerof water and a container of molasses. The act of stirring introduces shearingstresses in the fluid. The “thinner,” less viscous, water is easy to stir; the “thicker,”more viscous, molasses is harder to stir. The physical effect is represented bythe shear stresses applied to the “stirrer” by the fluid.
The concept of viscosity is293Hutton: Fundamentals ofFinite Element Analysis2948. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid MechanicsUⴢyUⴢhhx(b)(a)Figure 8.1(a) Moving plate separated by a fluid layer from a fixedsurface. (b) Velocity profile across the fluid thickness.embodied in Newton’s law of viscosity [2], which states that the shear stress in afluid is proportional to the velocity gradient.In a one-dimensional case, the velocity gradient and Newton’s law of viscosity can be described in reference to Figure 8.1a.
A long flat plate is movingwith velocity U̇ in the x direction and separated from a fixed surface located aty = 0 by a thin fluid film of thickness h. Experiments show that the fluid adheresto both surfaces, so that the fluid velocity at the fixed surface is zero, and at themoving plate, the fluid velocity is U̇ (this phenomenon is known as the no slipcondition). If pressure is constant throughout the fluid, the velocity distributionbetween the moving plate and the fixed surface is linear, as in Figure 8.1b, so thefluid velocity at any point is given byu̇( y) =yU̇h(8.1)To maintain the motion, a force in the direction of motion must be applied to theplate. The force is required to keep the plate in equilibrium, since the fluid exertsa friction force that opposes the motion.
It is known from experiments thatthe force per unit area (frictional shearing stress) required to maintain motion isproportional to velocity U̇ of the moving plate and inversely proportional to distance h. In general, the frictional shearing stress is described in Newton’s law ofviscosity as =du̇dy(8.2)where the proportionality constant is called the absolute viscosity of the fluid.Absolute viscosity (hereafter simply viscosity) is a fundamental material property of fluid media since, as shown by Equation 8.2, the ability of a fluid to support shearing stress depends directly on viscosity.The relative importance of viscosity effects leads to yet other subsets of fluidmechanics problems, as mentioned. Fluids that exhibit very little viscosity aretermed inviscid and shearing stresses are ignored; on the other hand, fluids withsignificant viscosity must be considered to have associated significant sheareffects.
To place the discussion in perspective, water is considered to be an incompressible, viscous fluid, whereas air is a highly compressible yet inviscidHutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.2 Governing Equations for Incompressible Flowfluid.
In general, liquids are most often treated as incompressible but the viscosity effects depend specifically on the fluid. Gases, on the other hand, are generally treated as compressible but inviscid.In this chapter, we examine only incompressible fluid flow. The mathematics and previous study required for examination of compressible flow analysis isdeemed beyond the scope of this text. We, however, introduce viscosity effects inthe context of two-dimensional flow and present the basic finite element formulation for solving such problems.
The extension to three-dimensional fluid flowis not necessarily as straightforward as in heat transfer and (as shown in Chapter 9) in solid mechanics. Our introduction to finite element analysis of fluid flowproblems shows that the concepts developed thus far in the text can indeed beapplied to fluid flow but, in the general case, the resulting equations, althoughalgebraic as expected from the finite element method, are nonlinear and specialsolution procedures must be applied.8.2 GOVERNING EQUATIONSFOR INCOMPRESSIBLE FLOWOne of the most important physical laws governing motion of any continuousmedium is the principle of conservation of mass.
The equation derived by application of this principle is known as the continuity equation. Figure 8.2 shows adifferential volume (a control volume) located at an arbitrary, fixed position ina three-dimensional fluid flow. With respect to a fixed set of Cartesian axes,the velocity components parallel to the x, y, and z axes are denoted u, v, and w,respectively.
(Note that here we take the standard convention of fluid mechanicsby denoting velocities without the “dot” notation.) The principle of conservationof mass requires that the time rate of change of mass within the volume mustv⫹w⫹u⫹dxuyw⭸vdy⭸ydy⭸wdz⭸zdzvxzFigure 8.2 Differential volume element inthree-dimensional flow.⭸udx⭸x295Hutton: Fundamentals ofFinite Element Analysis2968. Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid Mechanicsbe in balance with the net mass flow rate into the volume. Total mass inside thevolume is dV, and since dV is constant, we must have∂dV =(mass flow in − mass flow out)∂tand the partial derivative is used because density may vary in space as well astime.
Using the velocity components shown, the rate of change of mass in thecontrol volume resulting from flow in the x direction is∂ ( u)dx dy dzṁ x = u dy dz − u +(8.3a)∂xwhile the corresponding terms resulting from flow in the y and z directions are∂ ( v)ṁ y = v dx dz − v +dy dx dz(8.3b)∂y∂ ( w)ṁ z = w dx dy − w +dz dx dy(8.3c)∂zThe rate of change of mass then becomes∂ ( u)∂ ( v)∂ ( w)∂dV = ṁ x + ṁ y + ṁ z = −++dx dy dz∂t∂x∂y∂zNoting that dV = dx dy dz , Equation 8.4 can be written as∂∂∂∂u∂v∂w∂+u+v+w+++=0∂t∂x∂y∂z∂x∂y∂z(8.4)(8.5)Equation 8.5 is the continuity equation for a general three-dimensional flowexpressed in Cartesian coordinates.Restricting the discussion to steady flow (with respect to time) of an incompressible fluid, density is independent of time and spatial coordinates so Equation 8.5 becomes∂u∂v∂w++=0∂x∂y∂z(8.6)Equation 8.6 is the continuity equation for three-dimensional, incompressible,steady flow expressed in Cartesian coordinates.
As this is one of the most fundamental equations in fluid flow, we use it extensively in developing the finiteelement approach to fluid mechanics.8.2.1 Rotational and Irrotational FlowSimilar to rigid body dynamics, consideration must be given in fluid dynamicsas to whether the flow motion represents translation, rotation, or a combinationof the two types of motion.
Generally, in fluid mechanics, pure rotation (i.e.,Hutton: Fundamentals ofFinite Element Analysis8. Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.2 Governing Equations for Incompressible Flowttt ⫹ dtt ⫹ dt(b)(a)Figure 8.3 Fluid element in (a) rotational flow and (b) irrotational flow.rotation about a fixed point) is not of as much concern as in rigid body dynamics.Instead, we classify fluid motion as rotational (translation and rotation combined) or irrotational (translation only). Owing to the inherent deformability offluids, the definitions of translation and rotation are not quite the same as forrigid bodies. To understand the difference, we focus on the definition of rotationin regard to fluid flow.A flow field is said to be irrotational if a typical element of the moving fluidundergoes no net rotation.
A classic example often used to explain the concept isthat of the passenger carriages on a Ferris wheel. As the wheel turns through onerevolution, the carriages also move through a circular path but remain in fixedorientation relative to the gravitational field (assuming the passengers are wellbehaved).
As the carriage returns to the starting point, the angular orientation ofthe carriage is exactly the same as in the initial orientation, hence no net rotationoccurred. To relate the concept to fluid flow, we consider Figure 8.3, depictingtwo-dimensional flow through a conduit. Figure 8.3a shows an element of fluidundergoing rotational flow. Note that, in this instance, we depict the fluid element as behaving essentially as a solid. The fluid has clearly undergone translation and rotation.
Figure 8.3b depicts the same situation in the case of irrotationalflow. The element has deformed (angularly), and we indicate that angular deformation via the two angles depicted. If the sum of these two angles is zero, theflow is defined to be irrotational. As is shown in most basic fluid mechanics textbooks [2], the conditions for irrotationality in three-dimensional flow are∂v∂u−=0∂x∂y∂u∂w−=0∂x∂z(8.7)∂w∂v−=0∂y∂zWhen the expressions given by Equations 8.7 are not satisfied, the flow is rotational and the rotational rates can be defined in terms of the partial derivatives ofthe same equation. In this text, we consider only irrotational flows and do notproceed beyond the relations of Equation 8.7.297Hutton: Fundamentals ofFinite Element Analysis2988.
Applications in FluidMechanicsCHAPTER 8Text© The McGraw−HillCompanies, 2004Applications in Fluid Mechanics8.3 THE STREAM FUNCTION INTWO-DIMENSIONAL FLOWWe next consider the case of two-dimensional, steady, incompressible, irrotational flow. (Note that we implicitly assume that viscosity effects are negligible.)Applying these restrictions, the continuity equation is∂v∂u+=0∂x∂y(8.8)and the irrotationality conditions reduce to∂u∂v−=0∂y∂x(8.9)Equations 8.8 and 8.9 are satisfied if we introduce (define) the stream function(x , y) such that the velocity components are given byu=∂∂y∂v=−∂x(8.10)These velocity components automatically satisfy the continuity equation.
Theirrotationality condition, Equation 8.10, becomes∂v∂ ∂∂∂∂ 2∂u∂ 2−=−−=+= ∇ 2 = 0(8.11)2∂y∂x∂y ∂y∂x∂x∂x∂ y2Equation 8.11 is Laplace’s equation and occurs in the governing equations formany physical phenomena. The symbol ∇ represents the vector derivative operator defined, in general, by∇=∂∂∂i+j+k in Cartesian coordinates∂x∂y∂zand ∇ 2 = ∇ · ∇Let us now examine the physical significance of the stream function (x , y)in relation to the two-dimensional flow. In particular, we consider lines in the(x, y) plane (known as streamlines) along which the stream function is constant.If the stream function is constant, we can write∂∂d =dx +dy = 0(8.12)∂x∂yord = −v dx + u dy = 0(8.13)The tangent vector at any point on a streamline can be expressed asnt = dx i + dyj and the fluid velocity vector at the same point is V = ui + vj .Hence, the vector product V × nt = (−v dx + u dy)k has zero magnitude, perHutton: Fundamentals ofFinite Element Analysis8.
Applications in FluidMechanicsText© The McGraw−HillCompanies, 20048.3 The Stream Function in Two-Dimensional FlowEquation 8.13. The vector product of two nonzero vectors is zero only if the vectors are parallel. Therefore, at any point on a streamline, the fluid velocity vectoris tangent to the streamline.8.3.1 Finite Element FormulationDevelopment of finite element characteristics for fluid flow based on the streamfunction is straightforward, since (1) the stream function (x , y) is a scalarfunction from which the velocity vector components are derived by differentiation and (2) the governing equation is essentially the same as that for twodimensional heat conduction.
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