External flows are defined as those motions of a fluid medium which occur when the medium flows around a body immersed in it. External flows may be addressed by the methods of fluid mechanics.

In the general case, the motion of a fluid medium in a certain volume can be specified if, at any moment of time t, the fluid velocity field of the medium at any point of the volume can be determined (or calculated with the required accuracy). At present, this problem has been solved only for some particular cases. To a large extent, this difficulty is associated with defining boundary conditions. When a fluid medium flows around bodies immersed in a fluid, the zone of influence of the body on the flow can extend over a considerable distance both up and down the flow. Inside the fluid medium, sharp boundaries of the flow can arise (for instance, cavities in cavitation or shock waves in retarding supersonic flows). To describe the motion of the fluid medium, an appropriate mathematical model for this phenomenon must be chosen. As a rule, only the most essential properties of the medium are considered because the wider the context of the problem, the more difficult it is to close the mathematical model, to carry out physical analysis, and in the long run, to correlate theory with experiment. In many cases, it is the correct choice of a simple model that guarantees progress in the solution of a posed problem.

For gases, two models of fluid flow are used: molecular and viscous; for liquids, only a viscous model is used.

The molecular flow of gases is considered typical when the mean tree path length λ of molecules exceeds the characteristic linear dimension of a streamlined body d. The Knudsen Number, , is the dimensionless characteristic of the rarefied gas medium. The motion of gases at any Knudsen number is described by the Boltzmann equation. The equations of classical hydrodynamics follow the Boltzmann equation as the limiting case for the Knudsen number Kn → 0 .

The fundamental difference between the molecular and the viscous models of fluid flow lies in the statement of boundary conditions on the surface of streamlined bodies. Instead of conditions of equality of velocity and temperature of the gas at a solid wall (typical for viscous models), molecular models leave scope for a gas to slip and for the temperature to jump. In strongly-rarefied gases, the notions of temperature or velocity have to be interpreted differently than in the case of higher pressure systems.

Consider now the flow about a body at small Knudsen numbers. We assume that the continuous medium model can be applied to a fluid—which implies a continuous distribution of the substance and of the physical characteristics of its state and motion in space. An important property which differentiates a liquid or gaseous medium from a solid medium is their fluidity. Offering a noticeable counteraction to compressive strain, the fluids poorly resist shear deformation, i.e., a mutual slip of the layers of the medium.

The lower the tangential stresses that arise in motion, the less the relative velocity of mutual slip. This is what distinguishes fluids from granular solid media, whose slip is accompanied by the appearance of a friction force which is only slightly dependent on the slip velocity. There is practically no constant component of the resistance force in the fluid.

A complete set of equations which describes the motion of fluids, approximated as a continuous medium, is called the Navier–Stokes equations. An analysis of these equations reveals a set of dimensionless numbers or similarity criteria which describe external flow around a compact body with a geometrical shape and a length scale d. The description of the external flow includes the velocity field of fluid particles , temperature distribution T(, t) and density ρ(, t) of the medium in the entire volume under study.

Depending on a particular statement of the problem, the set of similarity criteria can be wider or narrower but, as a rule, it includes the Reynolds Number Re = ρvd/η, Mach Number Ma = v/a, the Prandtl Number Pr = ηCp/λ and the ratio of heat capacities at constant pressure and constant volume cp and cv (or an adiabatic exponent) γ = cp/cv. The notation used here is that v and a are the characteristic values of flow velocity and of sound velocity in the fluid while ρ, η, λ are density, viscosity and thermal conductivity, respectively. To this set of similarity criteria may sometimes be added the temperature factor, or the ratio of temperature on the wall (of the surface of a streamlined body) Tw to the temperature of a free-stream flow T. Accounting for the instability of the flow would require consideration of the Strouhal Number and accounting for mass or volume forces, the Froude Number.

The influence of the various similarity criteria on external flow varies from case to case. For a closer analyses of these influences, see the articles on Aerodynamics, Boundary Layers, Compressible Flow, Drag Coefficient, Inviscid Flow, Turbulent Flow, Free Convection, Jets and Waves in Fluids. Here, only the fundamental features of flows around bodies as a function of free-stream velocity are considered; to be more exact, these cases are governed by only two similarity criteria, i.e., the Reynolds and Mach numbers. It is interesting to note that there exist rather wide ranges of variations for both criteria, within which their particular values play no part in the description of flow. These are the self-similarity ranges of Reynolds numbers from 103 to 106 and of Mach numbers from 5 to ∞ (the numerical values are given for air flows). The discussion below focuses on flows with low values of Reynolds and Mach numbers. The assumption of a simple, axi-symmetric geometry (a sphere or a cylinder) as the shape of the streamlined bodies in all flows considered here does not detract from the general applicability of the concepts presented.

Visualization of a supersonic gas flow, with high values of Re and Ma, around the sphere (Figure 1) makes it possible to divide the entire field of flow into a number of areas for each of which a mathematical model of the local phenomena may be used.

In Figure 1, the uniform (homogeneous) incoming flow retains its free-stream velocity in the zone to the left of the bow shock wave 1. Conversely, at the right of jump 1, the flow undergoes considerable deformations, and the range of velocity variations is rather wide. In the area of the front stagnation point on the surface of body (2), the flow is subsonic and only on line 3 does the flow accelerate anew up to Ma = 1. This line is called the sound line and the point of its intersection with the body surface—sound point 4—is important for determining the intensity of heat transfer, both in laminar and in turbulent flows, in the Boundary Layer.

Flow round a cylinder or sphere.

Figure 1. Flow round a cylinder or sphere.

The sphere and the cylinder belong to a class of poorly-streamlined bodies. This means that the boundary layer leaves their surface at the mid- (or the largest) cross-section of the body, where the flow is accelerated to maximum velocity and local pressure p reaches a minimum (see Inviscid Flow). The separated or free boundary layer forms a shear flow 6, which separates the stagnant zone 7 in the bottom part of the body from an inviscid flow 5 zone expanding in Prandtl-Mayer waves.

When the free boundary layers 6 come together, a trailing vortex neck is formed behind the body which divides the entire trailing vortex into a short-range zone (denoted by B) and a long-range zone (denoted by C in Figure 1). The long-range trailing vortex (9) is a typical isobaric turbulent jet (see Jets), around which a suspended rear shock wave 8 is formed.

For a flow with high values of Mach and Reynolds numbers, the shock wave and the boundary layer can be treated as infinitely-thin, discontinuous areas, which allows the use of an inviscid flow model (see Inviscid Flow). In this case, the velocity field in the area A can be calculated adequately up to the separation point of the boundary layer.

It is well-known that in high-velocity gas flows, the stagnation temperature behind the shock wave can reach rather high values. This, in turn, brings about a considerable difference in the gas composition behind the shock wave and in the free-stream flow. Dissociation of molecules, ionization of atoms, numerous physicochemical interactions between the components of the gas mixture and the material of the streamlined body occur (see Ablation).

All these phenomena reflect the accuracy of predicting the thermodynamic and transfer properties of the medium (see, for instance, Thermal Conductivity), which must be known in order to apply the inviscid flow model. In particular, the adiabatic exponent γ = cp/cv varies considerably with increasing temperature in an air-type gas mixture so that the error in calculating the pressure may be quite large.

In this connection, a modified Newton formula for the calculation of local pressure on the surface of the body with an arbitrary geometry evokes interest because, in essence, it does not require any knowledge about the mechanisms of real physico-chemical processes behind the bow shock wave. When solving the problem involving the resistance of bodies in a fluid flow, Newton believed that all the fluid particles are the same, fill in the space ahead of the body and do not interact with one another. The particles strike against the body surface quite inelastically, i.e., the particles lose the momentum component normal to the body surface which is the cause of the surplus pressure exerted on the nose and the windwood parts of the body surface (Figure 2). As for body surface parts which find themselves in the wind shadow (dashed in Figure 2, Newton assumed that the gauge pressure is zero there.

Bases of Newton method for force on a body in a flow.

Figure 2. Bases of Newton method for force on a body in a flow.

If an element of the body surface with area dS is inclined to the velocity vector of the free-stream flow at angle θ, then the mass flow rate of the gas from which momentum is lost is ρVdS sin θ and the normal (“lost”) component of velocity is V sin θ. According to the law of momentum conservation, (p – p)dS = [ρVdS sin θ][V sin θ – 0], where the gauge pressure on the surface of the body with an angle of slope θ is determined by the relation called the Newton formula:

The Newton resistance model postulates that the direction of fluid particles motion in the flow around the body varies instantaneously. In the general case, this assumption is not fulfilled. Disturbances caused by the presence of the body propagate in a subsonic flow well upstream. In supersonic flows, however, the disturbance region is limited by a shock wave (Figure 1). At large Mach numbers (Ma → 5) , the thickness of the shock layer is small and the Newton scheme is realized more or less adequately. In the modern version, Newton’s formula is modified by replacing the denominator by the stagnation pressure (at Ma → 5 the difference of from does not exceed 5%).

From Newton’s formula for pressure distribution, the effect of the geometry of the body on drag coefficient can be defined easily (see Drag Coefficient); to be more precise, the formula gives only that part of the drag coefficient which is not related to viscous friction. For a spherical segment or a cylindrical sector with central angle ω (Figure 3) the respective expressions are:

Force on a spherical segment or cylindrical sector.

Figure 3. Force on a spherical segment or cylindrical sector.

Hence, the drag coefficient of the sphere (ω = π/2) is CD = 1; of the cylinder, CD = 4/3; and of the flat-end face perpendicular to the flow, CD = 2. For the cone with a half-angle at the apex θ, the drag coefficient is CD = 2 sin2 θ.

The Newton formula also permits defining the positions of sound points in the surface of bodies whose shapes have no breaks (i.e., of smooth bodies). Gas motion that follows the shock wave from the critical point along the surface obeys the laws of adiabatic expansion, when the local pressure p is related to the local Mach number as

Thus the ratio of the pressure p* on the sound line, to the stagnation pressure when Ma = 1, within a broad range of γ variation from 1.05 to 1.4, will be:

Since by the Newton formula the ratio is proportional to cos2 ω, the range of variation of the central angle ω* at the sound point is limited, ω* = 36° – 41°.

A slight dependence of the coordinate of the sound point on the adiabatic exponent γ allows the determination, with sufficient accuracy, of the distribution of the subsonic flow velocity in the vicinity of the stagnation point of a spherical or cylindrical blunt body. So, the velocity gradient on the longitudinal coordinate X, directed along the blunted body generatrix, can be calculated as follows:

This parameter determines the effect of the dimension of the body RN on heat transfer in a laminar boundary layer. Here, a is the velocity of sound at the stagnation temperature .

We will now consider a further limiting case of flow around a body, i.e., the region of low free-stream velocities, to be more precise, the region of low Mach numbers Ma ≤ 0.3 and low Reynolds numbers Re ≤ 1.

The condition of low subsonic flow velocity allows the assumption that variations of density at various points in space filled with a fluid are not significant, i.e., the fluid is practically incompressible. However, the subsonic range of velocities prevents, as was noted above, the flow from being subdivided into free-stream and distorted flows. Therefore, exact solutions of the Navier-Stokes equations have only been obtained for rather specific conditions, whose applied value is limited. Thus, the well-known Stokes solution has been obtained for a single sphere in an infinite medium. When a spherical particle moves near the wall or some other solid surface, the surface affects the structure of flow formed by the sphere. For instance, a correction for the resistance force has the form [1 + 0.56(d/h)], where d is the sphere diameter and h is the distance from the flat surface along which the particle moves.

Flows for which the Reynolds number is rather small are called creeping motions. Interesting data have been obtained as a result of studying the fall of spherical particles under gravity. This case of a free motion of a body has been more thoroughly investigated (see Drag Coefficient). The stages characteristic of a fluid motion around spherical particles, and the trajectories of the fall in spheres, can be separated as follows:

  1. At small Reynolds numbers (Re ≤ 0.5), a smooth continuous flow around spheres is observed. No vortices arise in the bottom region. The trajectories of particle fall are strictly vertical. This strictly corresponds with a laminar flow.

  2. At Reynolds numbers greater than 10, the formation of initially unstable and then stable vortices is observed, which adhere to the spheres. The trajectories of fall warp gradually and acquire an angle of slope to the vertical (transition streamline flow).

  3. At Reynolds numbers greater than 100, the vortices begin to separate and at Re 500, separated vortices become regular. Leaving the bottom part of the body alternately, the vortices turn alternately to the right and to the left. Such a sequence of vortices is called the Karman vortex street. The vortex street moves with a velocity less than the velocity of flow V incoming on the body. The trajectory of particle fall is a twisting curve (turbulent, streamlined flow). At this stage, the concept of fall velocity becomes arbitrary and the length of the trajectory can differ considerably with the height of the fall.

All the above values of characteristic Reynolds numbers depend on body shape and the degree of turbulence of the main flow in which motion takes place. In streamline flow over an assembly of particles, their motion can change considerably if the mean distance between the particles is less than some limiting distance. It is significant that the lesser the Reynolds number, the stronger is the interference of neighboring particles. Thus, the drag coefficient of spherical particles increases three times if Re = 2 × 102 and only 1.5 times if Re = 104 at the same interparticle distance 1/d = 5 (here, 1 is the distance between the centers of neighboring spherical particles).

For group motion of particles of various shapes, the problem of interaction becomes more involved, but reliable experimental data are not available.

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