Chapter 7
Ideal Flow
Abstract
When a constant-density fluid flows without rotation and the pressure is measured with respect to its local hydrostatic value, the field equation for fluid motion is linear and the pressure can be determined from a nonlinear algebraic (Bernoulli) equation. Under these ideal-flow conditions, a single scalar stream function or velocity potential may be used to describe the flow. Ideal flows around corners, near walls, and past simple obstacles may be constructed by superimposing fundamental solutions. In two dimensions these functions are naturally combined into a complex potential that satisfies the Cauchy-Riemann equations for a complex analytic function. The complex potential allows two important ideal-flow theorems to be proved, and it allows complicated flows to be analyzed via complex-variable mapping techniques. For axisymmetric and three-dimensional ideal flows the fundamental equations are still linear but the extension to complex functions is lost. In particular, unsteady three-dimensional potential flow can be used to determine the apparent (or added) mass of a fully submerged, arbitrarily moving sphere.
Keywords
Added mass; Apparent mass; Axisymmetric flow; Blasius theorem; Complex potential; Conformal mapping; Ideal flow; Irrotational flow; Kutta-Zhukhovsky theorem; Potential flow; Stream function; Three-dimensional flow; Velocity potential• To describe the formulation and limitations of ideal flow theory
• To illustrate the use of the stream function and the velocity potential in two-dimensional, axisymmetric, and three-dimensional flows
• To derive and present classical ideal flow results for flows past simple objects
7.1. Relevance of Irrotational Constant-Density Flow Theory
When a constant-density fluid flows without rotation, and pressure is measured relative to its local hydrostatic value (see Section 4.9), the equations of fluid motion in an inertial frame of reference, (4.7) and (4.38), simplify to:
(4.10, 7.1)
even though the fluid’s viscosity μ may be non-zero. These are the equations of ideal flow. They are useful for developing a first-cut understanding of nearly any macroscopic fluid flow, and are directly applicable to low-Mach-number irrotational flows of homogeneous fluids away from solid boundaries. Ideal flow theory has abundant applications in the exterior aero- and hydrodynamics of moderate- to large-scale objects at non-trivial subsonic speeds. Here, moderate size (L) and non-trivial speed (U) are determined jointly by the requirement that the Reynolds number, Re = ρUL/μ (4.103), be large enough (typically Re ∼ 103 or greater) so that the combined influence of fluid viscosity and fluid element rotation is confined to thin layers on solid surfaces, commonly known as boundary layers.
The conditions necessary for the application of ideal flow theory are commonly present on the upstream side of many ordinary objects, and may even persist to the downstream side of some. Ideal flow analysis can predict fluid velocity away from solid surfaces, surface-normal pressure forces (when the boundary layer on the surface is thin and attached), acoustic streamlines, flow patterns that minimize form drag, and unsteady-flow fluid-inertia effects. Ideal flow theory does not predict viscous effects like skin friction or energy dissipation, so it is not directly applicable to interior flows in pipes and ducts, to boundary-layer flows, or to any rotational flow region. This final specification excludes low-Re flows and regions of turbulence.
Because (7.1) involves only first-order spatial derivatives, ideal flows only satisfy the no-through-flow boundary condition on solid surfaces. The no-slip boundary condition (4.94) is not applied in ideal flows, so non-zero tangential velocity at a solid surface may exist (Figure 7.1a). In contrast, a real fluid with a non-zero shear viscosity must satisfy the no-slip boundary condition (4.94) because (4.38) contains second-order spatial derivatives. At sufficiently high Re, there are two primary differences between ideal and real flows over the same object. First, viscous boundary layers containing rotational fluid form on solid surfaces in the real flow, and the thickness of such boundary layers, within which viscous diffusion of vorticity is important, approaches zero as Re → ∞ (Figure 7.1b). The second difference is the possible formation in the real flow of separated flow or wake regions that occur when boundary layers leave the surface on which they have developed to create a wider zone of rotational flow (Figure 7.2). Ideal flow theory is not directly applicable to such layers or regions of rotational flow. However, rotational flow regions may be easy to anticipate or identify, and may represent a small fraction of a total flow field so that predictions from ideal flow theory may remain worthwhile even when viscous flow phenomena are present. Further discussion of viscous-flow phenomena is provided in Chapters 9 and 10.
Figure 7.1 Comparison of a completely irrotational constant-density (ideal) flow (a) and a high Reynolds number flow (b). In both cases the no-through-flow boundary condition is applied. However, the ideal flow is effectively inviscid and the fluid velocity is tangent to and non-zero on the body surface. The high-Re flow includes thin boundary layers where fluid rotation and viscous effects are prevalent, and the no-slip boundary condition is enforced, but the velocity above the thin boundary layer is similar to that in the ideal flow.
For (4.10) and (7.1) to apply, fluid density ρ must be constant and the flow must be irrotational. If the flow is merely incompressible and contains baroclinic density variations, (4.10) will still be satisfied but (7.1) will not; it will need a body-force term like that in (4.84) and the reference pressure would have to be redefined. If the fluid is a homogeneous compressible gas with sound speed c, the constant density requirement will be satisfied when the Mach number, M = U/c (4.111), of the flow is much less than unity. The irrotationality condition is satisfied when fluid elements enter the flow field of interest without rotation and do not acquire any while they reside in it. Based on Kelvin’s circulation theorem (5.11) for constant density flow, this is possible when the body force is conservative and the net viscous torque on a fluid element is zero. Thus, a fluid element that is initially irrotational is likely to stay that way unless it enters a boundary layer, wake, or separated flow region where it acquires rotation via viscous diffusion. So, when initially irrotational fluid flows over a solid object, ideal flow theory most-readily applies to the outer region of the flow away from the object’s surface(s) where the flow is irrotational. Viscous flow theory is needed in the inner region where viscous diffusion of vorticity is important. Often, at high Re, the outer flow can be approximately predicted by ignoring the existence of viscous boundary layers. With this outer flow prediction, viscous flow equations can be solved for the boundary-layer flow and, under the right conditions, the two solutions can be adjusted until they match in a suitable region of overlap. This approach works well for objects like thin airfoils at low angles of attack when boundary layers remain thin and stay attached all the way to the foil’s trailing edge (Figure 7.1b). However, it is not satisfactory when the solid object has such a shape that one or more boundary layers separate from its surface before reaching its downstream edge (Figure 7.2), giving rise to a rotational wake flow or region of separated flow (sometimes called a separation bubble) that is not necessarily thin, no matter how high the Reynolds number. In this case, the limit of a real flow as μ → 0 
does not approach that of an ideal flow (μ = 0). Yet, upstream of boundary-layer separation, ideal flow theory may still provide a good approximation of the real flow.
does not approach that of an ideal flow (μ = 0). Yet, upstream of boundary-layer separation, ideal flow theory may still provide a good approximation of the real flow.
Figure 7.2 Schematic drawings of flows with boundary-layer separation. (a) Real flow past a cylinder where the boundary layers on the top and bottom of the cylinder leave the surface near its widest point. (b) Real flow past a sharp-cornered obstacle where the boundary layer leaves the surface at the corner. Upstream of the point of separation, ideal flow theory is usually a good approximation of the real flow.
In summary, the theory presented here does not apply to inhomogeneous fluids, high subsonic or supersonic flow speeds, boundary-layer flows, wake flows, interior flows, or any flow region where fluid elements rotate. However, the remaining flow possibilities are abundant, and include those that are commercially valuable (flight), naturally important (water waves), or readily encountered in our everyday lives (flow around vehicles, also see Chapters 8 and 14 for further examples).
Steady and unsteady irrotational constant-density flow fields around simple objects and through simple geometries in two and three dimensions are the subjects of this chapter. All the coordinate systems presented in Figure 3.3 are utilized herein.
Example 7.1
If the pressure is constant in an ideal flow, what does this imply about the fluid velocity?
Solution
When the pressure is constant, ∇ p 
is zero so (7.1) simplifies to Du/Dt = 0. Thus, the fluid velocity is constant following fluid particles, and each fluid particle retains the velocity U(xo,to) it had at a reference time to and reference location xo. If a moving coordinate system is attached to any fluid particle, then that coordinate system will be inertial and the fluid particle of interest will remain stationary at the origin of these moving coordinates. The flow about such a stationary point can only be composed of straining motion and rotation (3.10, 3.19), but there can be no rotation in an ideal flow. Thus, the most general possible ideal flow field about the origin in the moving coordinate system is straining flow. However, in this moving frame, the origin of coordinates is a stagnation point, a pressure maximum (see Example 7.8). Thus, to achieve constant pressure, the fluid velocity in the moving frame must be zero everywhere, which implies constant velocity in the original stationary frame of reference. Thus, constant pressure in a steady ideal flow implies constant fluid velocity.
is zero so (7.1) simplifies to Du/Dt = 0. Thus, the fluid velocity is constant following fluid particles, and each fluid particle retains the velocity U(xo,to) it had at a reference time to and reference location xo. If a moving coordinate system is attached to any fluid particle, then that coordinate system will be inertial and the fluid particle of interest will remain stationary at the origin of these moving coordinates. The flow about such a stationary point can only be composed of straining motion and rotation (3.10, 3.19), but there can be no rotation in an ideal flow. Thus, the most general possible ideal flow field about the origin in the moving coordinate system is straining flow. However, in this moving frame, the origin of coordinates is a stagnation point, a pressure maximum (see Example 7.8). Thus, to achieve constant pressure, the fluid velocity in the moving frame must be zero everywhere, which implies constant velocity in the original stationary frame of reference. Thus, constant pressure in a steady ideal flow implies constant fluid velocity.