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āWhen a flow is both frictionless and irrotational, pleasant things happen.ā We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer.. We can define a potential function ( x ļ¬ z ļ¬ t ) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum , assuming incompressible, inviscid and irrotational flow.
The Stream Function Is a clever device which allows us to wipe out the continuity equation and solve the momentum equation directly for the single variable. Continuity equation
The vorticity, or curl V, is an interesting function Where One important application is inviscid irrotational flow in the xy plane, where ĻZ = 0 and the above Eq. is reduced to This is the second-order Laplace equation for which many solutions and analytical techniques are known. Also, boundary conditions like Eq. reduce to
Geometric Interpretation of Stream Function The fancy mathematics above would serve by itself to make the stream function immortal and always useful to engineers. Even better, though, stream function has a beautiful geometric interpretation: Lines of constant stream function are streamlines of the flow. This can be shown as follows. The definition of a streamline in two-dimensional flow is By the definition of stream function we obtain
Geometric Interpretation of Stream Function Sign convention for flow in terms of change in stream function: (a) flow to the right if ĻU is greater; (b) flow to the left if ĻL is greater. Thus the change in Ļ across the element is numerically equal to the volume flow through the element. The volume flow between any two points in the flow field is equal to the change in stream function between those points:
Geometric Interpretation of Stream Function Example: If a stream function exists for the velocity field of find it, plot it, and interpret it.
Frictionless Irrotational Flow When a flow is both frictionless and irrotational, pleasant things happen. First, the momentum equation reduces to Eulerās equation Second, there is a great simplification in the acceleration term. A beautiful vector identity exists for the second term
Frictionless Irrotational Flow Divide by Ļ, and rearrange on the left-hand side. The entire equation into an arbitrary vector displacement dr: Nothing works right unless we can get rid of the third term. We want This will be true under various conditions:
- V is zero; trivial, no flow (hydrostatics).
- Ī¾ is zero; irrotational flow.
- dr is perpendicular to (Ī¾ x V) ; this is rather specialized and rare.
- dr is parallel to V; we integrate along a streamline
Velocity Potential Irrotationality gives rise to a scalar function similar and complementary to the stream Function. A vector with zero curl must be the gradient of a scalar function where Ļ =Ļ (x, y, z, t) is called the velocity potential function. Knowledge of Ļ thus immediately gives the velocity components Note that Ļ, unlike the stream function, is fully three-dimensional and not limited to two coordinates. It reduces a velocity problem with three unknowns u, v, and w to a single unknown potential Ļ. The velocity potential also simplifies the unsteady Bernoulli equation because if Ļ exists, we obtain
Velocity Potential Unsteady Bernoulli Equation then becomes a relation between Ļ and Ļ This is the unsteady irrotational Bernoulli equation. It is very important in the analysis of accelerating flow fields Orthogonality of Streamlines and Potential Lines If a flow is both irrotational and described by only two coordinates, Ļ and Ļ both exist and the streamlines and potential lines are everywhere mutually perpendicular except at a stagnation point. For example, for incompressible flow in the xy plane, we would have Cauchy-Riemann equations
Velocity Potential Generation of Rotationality A fluid flow which is initially irrotational may become rotational if
- High viscous forces induced by jets, wakes, or solid boundaries. In this situation Bernoulliās equation will not be valid in such viscous regions.
- There are entropy gradients caused by curved shock waves.
- There are density gradients caused by stratification (uneven heating) rather than by pressure gradients.
- There are significant noninertial effects such as the earthās rotation (the Coriolis acceleration).
Velocity Potential Generation of Rotationality A fluid flow which is initially irrotational may become rotational if Typical flow patterns illustrating viscous regions patched into nearly frictionless regions: (a) low subsonic flow past a body(U << a); frictionless, irrotational potential flow outside the boundary layer (Bernoulli and Laplace equations valid)
Velocity Potential Generation of Rotationality A fluid flow which is initially irrotational may become rotational if
- Internal flows, such as pipes and ducts, are mostly viscous, and the wall layers grow to meet in the core of the duct. Bernoulliās equation does not hold in such flows unless it is modified for viscous losses.
- External flows, such as a body immersed in a stream, are partly viscous and partly inviscid, the two regions being patched together at the edge of the shear layer or boundary layer.
Velocity Potential Generation of Rotationality A fluid flow which is initially irrotational may become rotational if The approach stream is irrotational; i.e., the curl of a constant is zero, but viscous stresses create a rotational shear layer beside and downstream of the body. The shear layer is laminar, or smooth, near the front of the body and turbulent, or disorderly, toward the rear. A separated, region usually occurs near the trailing edge, followed by an unsteady turbulent wake extending far downstream. Some sort of laminar or turbulent viscous theory must be applied to these viscous regions; they are then patched onto the outer flow, which is frictionless and irrotational. If the stream Mach number is less than about 0.3, the fluid flow is consider incompressible.
