Technology & Engineering
Inviscid Flow
Inviscid flow refers to the theoretical flow of a fluid with zero viscosity, meaning that there is no internal friction or resistance to the motion of the fluid. Inviscid flow is often used in the study of fluid dynamics to simplify calculations and understand the fundamental behavior of fluids. While real fluids have viscosity, the concept of inviscid flow helps in developing theoretical models for practical applications.
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eBook - PDFBasic Aerodynamics
Incompressible Flow
- Gary A. Flandro, Howard M. McMahon, Robert L. Roach(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
309 8 Viscous Incompressible Flow External aerodynamics was a disturbingly mysterious subject before Prandtl solved the mystery with his work on boundary layer theory from 1904 onwards. L. Rosenhead Laminar Boundary Layers, Oxford 1963 8.1 Introduction This chapter examines the role of viscosity in the flow of fluids and gases. Although the viscosity of air is small, it must be included in a flow model if we are to explain wing stall and frictional drag, for example. The four preceding chapters are con- cerned with the analysis of airfoils, wings, and bodies of revolution based on an assumption of Inviscid Flow (i.e., negligible viscous effects). The inviscid-flow model allowed analytical solutions to be developed for predicting, with satisfactory accu- racy, the pressure distribution on bodies of small-thickness ratio at a modest (or zero) angle of attack. However, the inviscid-flow model leads to results that are at odds with experience, such as the prediction that the drag of two-dimensional air- foils and right-circular cylinders is zero. This contradiction is resolved by realizing that actual flows exhibit viscous effects. Viscosity is discussed from a physical viewpoint in Chapter 2. In Chapters 5, 6, and 7, the existence of viscosity is acknowledged when it is necessary to advance an analytical derivation for an Inviscid Flow. Also, viscous effects are called on, with words like viscous drag and separation, when comparing the predicted and observed behavior of airfoils and wings. However, no analysis in this textbook has been developed thus far that provides the required detailed physical basis for these effects . The focus of this chapter is a detailed study of the role of viscosity in an incom- pressible flow, particularly regarding modifications to the behavior of airfoils and wings that was predicted based on an inviscid-flow model.
eBook - PDF- Joseph Katz(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
5 Viscous Incompressible Flow: Exact Solutions (Leading to Some Practical Engineering Solutions) 5.1 Introduction In Chapter 3 the effects of pressure in a fluid were isolated (because the fluid was not moving), and in Chapter 4 the inertia terms were added. The inclusion of viscosity, its effects, and the resulting velocity distribution are discussed here. For example, the velocity distribution for the laminar flow inside a pipe is formulated and the average velocity is calculated. This provides the relation between the simple 1D average velocity model (of Chapter 4) and the more complex (and realistic) 2D or 3D flows. The solutions presented early in this chapter are often called exact solutions . This means that, for a few limited cases, a set of logical assumptions leads to simplifi-cation of the fluid dynamic equations, which allows their solution (for laminar flow)! Also, the cases presented in this chapter (e.g., the flow in pipes) is often termed as internal flows. The discussion on external flows is delayed to the following three chapters. The second part of this chapter demonstrates the approach that evolved during the past 200 years for solving fluid dynamic problems (because there is no closed-form analytic solution to the complete fluid dynamic equations). According to this approach, to develop a practical engineering solution, we must start with a simple but exact solution that determines the major parameters and the basic trends of the problem (e.g., the pressure drop in a circular pipe versus the Reynolds number). Based on these parameters, an empirical database can be developed for treating a wider range of engineering problems. As an example, the viscous laminar flow model in circular pipes is extended into the high-Reynolds-number range and the effects of turbulent flow are discussed.
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
12 Inviscid FlowIn real-life situations, most flow is turbulent. In pipe flow, in open-channel flow, and in flow over immersed bodies, laminar conditions exist only rarely. To analyze problems in these areas of study, it is most beneficial to be able to obtain a velocity or a velocity distribution for the problem at hand. As we saw in the last chapter, however, solution of the Navier–Stokes equations for turbulent flow problems is no easy task. Exact solutions were obtainable only for simplified laminar flow cases. Consequently, it is often necessary to make approximations so that the engineer can formulate a working solution to a number of important problems.Let us for the moment consider flow past an object, as illustrated in Figure 12.1 . Streamlines of flow about the object also appear in the diagram. Upstream the flow is uniform. In the vicinity of the object, the flow pattern is altered from the uniform incoming flow—the object displaces the flow. Far from the surface, however, the fluid is not affected by the presence of an object. In the regions labeled A in Figure 12.1 , the streamlines are therefore uniform and parallel. At the surface, the fluid adheres because of friction. Regardless of how small the viscosity of the fluid is, velocity at the wall is zero. We therefore conclude that viscous effects can be neglected in a flow field except in the vicinity of a surface. It is thus possible to divide the flow field into two regions. The first of these is a nonviscous region where viscosity need not be included in the equations of motion. The second is a viscous region for the fluid in the immediate vicinity of the object.FIGURE 12.1 Streamlines of flow past an object.A study of flow of a nonviscous or inviscid fluid is the topic of interest in this chapter. Flow in the vicinity of an immersed object or near a surface is often referred to as boundary-layer flow , the topic of the following chapter. In this chapter, we will develop equations for steady, incompressible, Inviscid Flow. The continuity and Euler equations will be derived from the Navier–Stokes equations of Chapter 11
eBook - PDFAerodynamics of Road Vehicles
From Fluid Mechanics to Vehicle Engineering
- Wolf-Heinrich Hucho(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
2.2.2 Internal flow Internal flow is that which is surrounded by walls. In the simple case of Fig. 2.3 all streamlines are parallel to the pipe axis. In general, internal flows cannot be divided into an Inviscid Flow far away from the walls and a 0 = 2/? I L Figure 2.3 Velocity distribution of the flow through a pipe viscous boundary-layer flow close to the walls. The effects of viscosity are found everywhere in the flow field. The development of an internal viscous flow is again characterized by the Reynolds number V D Reu = ^ψ-(2-5) based on a velocity typical for the problem, e.g. the mean velocity V m as in Fig. 2.3, and the pipe diameter D as a typical length. For different values of Re O , different types of flow may occur. 2.3 External flow problems 2.3.1 Basic equations for inviscid incompressible flow The development of the Inviscid Flow at the outer edge of the boundary layer determines the pressure distribution on the body surface. Therefore the fundamentals of such a flow are discussed first. To begin with, the law of mass conservation has to be formulated. The most simple form of this law is for incompressible flow (p = constant): ws = constant (2.6) where s denotes the local cross-section of a small stream-tube as in Fig. 2.2 and w is the local velocity, which is assumed to be constant across s. Eqn 2.6 indicates narrow distances between the streamlines in regions of high velocity and vice versa, see Fig. 1.1. Furthermore the flow obeys Newton's well-known law of momentum conservation: mass times acceleration is equal to the sum of the acting External flow problems 51 forces. If this law is applied to an Inviscid Flow, it turns out that inertia forces and pressure forces are balanced.
eBook - PDF- Frank H. Berkshire, Simon J. A. Malham, J. Trevor Stuart(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
Part I Inviscid Flow 1 Flow and Transport 1.1 Fluids and the Continuum Hypothesis A material exhibits flow if shear forces, however small, lead to a deformation which is unbounded – we could use this as a definition of a fluid. A solid has a fixed shape, or at least a strong limitation on its deformation when force is applied to it. Within the category of ‘fluids’, we include liquids and gases. The main distinguishing feature between these two fluids is the notion of compressibility. Gases are usually compressible – as we know from everyday aerosols. Liquids are generally incompressible – a feature essential to all modern car brakes. However, some gas flows can also be incompressible, particularly at low speeds. Fluids can be further subcategorised. There are ideal or inviscid fluids. In such fluids, the only internal force present is pressure, which acts so that fluid flows from a region of high pressure to one of low pressure. The equations for an ideal fluid have been applied to wing and aircraft design (as a limit of high Reynolds-number flow). However, fluids can exhibit internal frictional forces which model a ‘stickiness’ property of the fluid which involves energy loss – these are known as viscous fluids. Some fluids/material known as ‘non-Newtonian or complex fluids’ exhibit even stranger behaviour, their reaction to deformation may depend on: (i) past history (earlier deformations), for example some paints; (ii) temperature, for example some polymers or glass; (iii) the size of the deformation, for example some plastics or silly putty. For any real fluid there are three natural length scales: 1. L molecular , the molecular scale characterised by the mean-free-path distance of molecules between collisions; 2. L fluid , the medium scale of a fluid parcel, the fluid droplet in the pipe or ocean flow; 3. L macro , the macro-scale which is the scale of the fluid geometry, the scale of the container the fluid is in, whether a beaker or an ocean.
eBook - PDF- Bhimsen K. Shivamoggi(Author)
- 2011(Publication Date)
- Wiley-Interscience(Publisher)
314 Dynamics of In viscid, Compressible Fluid Flows EXERCISES 1. Estimate the error incurred in writing the boundary condition (46). 2. Obtain the solution for a supersonic flow past a cone by starting with the assumption that the flow is conical (and not using the slender axisymmetric body theory). 3. Linearize the transonic-flow equation using the hodograph transformation, and investigate the prospects of its solvability. 4. Set up the boundary-value problem for a nonlinear axisymmetric flow past a cone, and investigate its prospects of solvability. 4 DYNAMICS OF VISCOUS FLUID FLOWS 4.1. Exact Solutions to Equations of Viscous Fluid Flows The mathematical theory of ideal-fluid flow given so far provides a powerful approach to the solution of several problems and gives satisfactory descriptions of such characteristics of flows of the real fluid as (a) the main characteristics of wave motion and (b) the pressure field on streamlined bodies placed in flows. However, this theory is unable to indicate how nearly the flow field (the whole or part of it) will be irrotational. Therefore, application of the results from this theory requires the clarification of the circumstances in which the ideal-fluid assumption is valid. Basically, the ideal-fluid assumption is useful insofar as it may describe the behavior of a real fluid in the limit of vanishing viscosity. However, because of the contamination by vorticity within a boundary layer near a solid surface, the ideal-fluid theory does not give a correct description of the flows near solid boundaries and cannot describe such things as skin friction and form drag of a body placed in a flow. In order to remove these discrepancies, an understanding and inclusion of the effects of viscosity of a real fluid is essential. Among the effects produced by the fluid viscosity are (1) generation of shearing stresses in the fluid; (2) maintenance of a zero slip-velocity of the fluid at a solid boundary.
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