Stream function

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In fluid dynamics, two types of stream function are defined:

Streamlines – lines with a constant value of the stream function – for the incompressible potential flow around a circular cylinder in a uniform onflow.

The properties of stream functions make them useful for analyzing and graphically illustrating flows.

The remainder of this article describes the two-dimensional stream function.

Two-dimensional stream function

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Assumptions

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The two-dimensional stream function is based on the following assumptions:

  • The space domain is three-dimensional.
  • The flow field can be described as two-dimensional plane flow, with velocity vector
 
 

Although in principle the stream function doesn't require the use of a particular coordinate system, for convenience the description presented here uses a right-handed Cartesian coordinate system with coordinates  .

Derivation

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The test surface

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Consider two points   and   in the   plane, and a curve  , also in the   plane, that connects them. Then every point on the curve   has   coordinate  . Let the total length of the curve   be  .

Suppose a ribbon-shaped surface is created by extending the curve   upward to the horizontal plane    , where   is the thickness of the flow. Then the surface has length  , width  , and area  . Call this the test surface.

Flux through the test surface

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The volume flux through the test surface connecting the points   and  

The total volumetric flux through the test surface is

 

where   is an arc-length parameter defined on the curve  , with   at the point   and   at the point  . Here   is the unit vector perpendicular to the test surface, i.e.,

 

where   is the   rotation matrix corresponding to a   anticlockwise rotation about the positive   axis:

 

The integrand in the expression for   is independent of  , so the outer integral can be evaluated to yield

 

Classical definition

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Lamb and Batchelor define the stream function   as follows.[3]

 

Using the expression derived above for the total volumetric flux,  , this can be written as

 .

In words, the stream function   is the volumetric flux through the test surface per unit thickness, where thickness is measured perpendicular to the plane of flow.

The point   is a reference point that defines where the stream function is identically zero. Its position is chosen more or less arbitrarily and, once chosen, typically remains fixed.

An infinitesimal shift   in the position of point   results in the following change of the stream function:

 .

From the exact differential

 

so the flow velocity components in relation to the stream function   must be

 

Notice that the stream function is linear in the velocity. Consequently if two incompressible flow fields are superimposed, then the stream function of the resultant flow field is the algebraic sum of the stream functions of the two original fields.

Effect of shift in position of reference point

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Consider a shift in the position of the reference point, say from   to  . Let   denote the stream function relative to the shifted reference point  :

 

Then the stream function is shifted by

 

which implies the following:

  • A shift in the position of the reference point effectively adds a constant (for steady flow) or a function solely of time (for nonsteady flow) to the stream function   at every point  .
  • The shift in the stream function,  , is equal to the total volumetric flux, per unit thickness, through the surface that extends from point   to point  . Consequently   if and only if   and   lie on the same streamline.

In terms of vector rotation

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The velocity   can be expressed in terms of the stream function   as

 

where   is the   rotation matrix corresponding to a   anticlockwise rotation about the positive   axis. Solving the above equation for   produces the equivalent form

 

From these forms it is immediately evident that the vectors   and   are

  • perpendicular:  
  • of the same length:  .

Additionally, the compactness of the rotation form facilitates manipulations (e.g., see Condition of existence).

In terms of vector potential and stream surfaces

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Using the stream function, one can express the velocity in terms of the vector potential  

 

where  , and   is the unit vector pointing in the positive   direction. This can also be written as the vector cross product

 

where we've used the vector calculus identity

 

Noting that  , and defining  , one can express the velocity field as

 

This form shows that the level surfaces of   and the level surfaces of   (i.e., horizontal planes) form a system of orthogonal stream surfaces.

Alternative (opposite sign) definition

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An alternative definition, sometimes used in meteorology and oceanography, is

 

Relation to vorticity

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In two-dimensional plane flow, the vorticity vector, defined as  , reduces to  , where

 

or

 

These are forms of Poisson's equation.

Relation to streamlines

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Consider two-dimensional plane flow with two infinitesimally close points   and   lying in the same horizontal plane. From calculus, the corresponding infinitesimal difference between the values of the stream function at the two points is

 

Suppose   takes the same value, say  , at the two points   and  . Then this gives

 

implying that the vector   is normal to the surface  . Because   everywhere (e.g., see In terms of vector rotation), each streamline corresponds to the intersection of a particular stream surface and a particular horizontal plane. Consequently, in three dimensions, unambiguous identification of any particular streamline requires that one specify corresponding values of both the stream function and the elevation (  coordinate).

The development here assumes the space domain is three-dimensional. The concept of stream function can also be developed in the context of a two-dimensional space domain. In that case level sets of the stream function are curves rather than surfaces, and streamlines are level curves of the stream function. Consequently, in two dimensions, unambiguous identification of any particular streamline requires that one specify the corresponding value of the stream function only.

Condition of existence

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It's straightforward to show that for two-dimensional plane flow   satisfies the curl-divergence equation

 

where   is the   rotation matrix corresponding to a   anticlockwise rotation about the positive   axis. This equation holds regardless of whether or not the flow is incompressible.

If the flow is incompressible (i.e.,  ), then the curl-divergence equation gives

 .

Then by Stokes' theorem the line integral of   over every closed loop vanishes

 

Hence, the line integral of   is path-independent. Finally, by the converse of the gradient theorem, a scalar function   exists such that

 .

Here   represents the stream function.

Conversely, if the stream function exists, then  . Substituting this result into the curl-divergence equation yields   (i.e., the flow is incompressible).

In summary, the stream function for two-dimensional plane flow exists if and only if the flow is incompressible.

Potential flow

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For two-dimensional potential flow, streamlines are perpendicular to equipotential lines. Taken together with the velocity potential, the stream function may be used to derive a complex potential. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.

Summary of properties

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The basic properties of two-dimensional stream functions can be summarized as follows:

  1. The x- and y-components of the flow velocity at a given point are given by the partial derivatives of the stream function at that point.
  2. The value of the stream function is constant along every streamline (streamlines represent the trajectories of particles in steady flow). That is, in two dimensions each streamline is a level curve of the stream function.
  3. The difference between the stream function values at any two points gives the volumetric flux through the vertical surface that connects the two points.

Two-dimensional stream function for flows with time-invariant density

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If the fluid density is time-invariant at all points within the flow, i.e.,

 ,

then the continuity equation (e.g., see Continuity equation#Fluid dynamics) for two-dimensional plane flow becomes

 

In this case the stream function   is defined such that

 

and represents the mass flux (rather than volumetric flux) per unit thickness through the test surface.

See also

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References

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Citations

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  1. ^ Lagrange, J.-L. (1868), "Mémoire sur la théorie du mouvement des fluides (in: Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, année 1781)", Oevres de Lagrange, vol. Tome IV, pp. 695–748
  2. ^ Stokes, G.G. (1842), "On the steady motion of incompressible fluids", Transactions of the Cambridge Philosophical Society, 7: 439–453, Bibcode:1848TCaPS...7..439S
    Reprinted in: Stokes, G.G. (1880), Mathematical and Physical Papers, Volume I, Cambridge University Press, pp. 1–16
  3. ^ Lamb (1932, pp. 62–63) and Batchelor (1967, pp. 75–79)

Sources

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