4.3 Stream function

Stream functions are mathematical tools used in fluid dynamics to describe and analyze fluid flow in two or three dimensions. They simplify the analysis of incompressible and irrotational flows by reducing the number of dependent variables and automatically satisfying the continuity equation.

In two-dimensional flow, stream functions relate to velocity components and remain constant along streamlines. This property allows for easy visualization of flow patterns and calculation of flow rates. In three-dimensional flow, stream functions become vector quantities, providing similar benefits for analyzing complex flow structures.

Definition of stream function

• Stream function is a mathematical tool used in fluid dynamics to describe and analyze the flow of fluids in two or three dimensions
• Denoted by the Greek letter $\psi$ (psi), stream function is a scalar function that varies in space and time
• Stream function is particularly useful for incompressible and irrotational flows, where it can simplify the analysis of flow patterns and properties

Stream function in two-dimensional flow

Relationship between stream function and velocity components

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• In two-dimensional flow, stream function is related to the velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively
• The velocity components can be expressed in terms of the partial derivatives of the stream function:
  • $u = \frac{\partial \psi}{\partial y}$
  • $v = -\frac{\partial \psi}{\partial x}$
• These relationships ensure that the continuity equation for incompressible flow $(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0)$ is automatically satisfied

Constant value of stream function along streamlines

• In two-dimensional flow, streamlines are lines that are everywhere tangent to the velocity vector at a given instant
• The value of the stream function remains constant along a streamline
• This property allows for the visualization of flow patterns by plotting contours of constant stream function values
• The difference in stream function values between two streamlines represents the volumetric flow rate per unit depth between those streamlines

Stream function in three-dimensional flow

Relationship between stream function and velocity components

• In three-dimensional flow, stream function is a vector quantity denoted by $\vec{\psi}$
• The velocity components $u$, $v$, and $w$ in the $x$, $y$, and $z$ directions, respectively, can be expressed in terms of the stream function components:
  • $u = \frac{\partial \psi_z}{\partial y} - \frac{\partial \psi_y}{\partial z}$
  • $v = \frac{\partial \psi_x}{\partial z} - \frac{\partial \psi_z}{\partial x}$
  • $w = \frac{\partial \psi_y}{\partial x} - \frac{\partial \psi_x}{\partial y}$
• These relationships ensure that the continuity equation for incompressible flow $(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0)$ is automatically satisfied

Constant value of stream function along stream surfaces

• In three-dimensional flow, stream surfaces are surfaces formed by the streamlines passing through a closed curve
• The value of the stream function remains constant along a stream surface
• The difference in stream function values between two stream surfaces represents the volumetric flow rate between those surfaces

Properties of stream function

Orthogonality of stream function and velocity potential

• In irrotational flows, where the vorticity is zero $(\nabla \times \vec{V} = 0)$, the stream function is orthogonal to the velocity potential $\phi$
• The velocity potential is another scalar function that describes the flow, and its gradient gives the velocity vector: $\vec{V} = \nabla \phi$
• The orthogonality of stream function and velocity potential means that their contours intersect at right angles

Laplace's equation for stream function

• In two-dimensional, incompressible, and irrotational flows, the stream function satisfies Laplace's equation:
  • $\nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = 0$
• This property allows for the use of analytical and numerical methods developed for solving Laplace's equation to determine the stream function

Determination of stream function

Calculation of stream function from velocity field

• If the velocity field is known, the stream function can be calculated by integrating the velocity components
• In two-dimensional flow, the stream function can be obtained by integrating the velocity components:
  • $\psi(x, y) = \int_0^y u(x, y') dy' + f(x)$ or $\psi(x, y) = -\int_0^x v(x', y) dx' + g(y)$
  • where $f(x)$ and $g(y)$ are arbitrary functions determined by boundary conditions
• In three-dimensional flow, the stream function components can be calculated using similar integration techniques

Boundary conditions for stream function

• To uniquely determine the stream function, appropriate boundary conditions must be specified
• Common boundary conditions include:
  • Specifying the value of the stream function along solid boundaries (e.g., $\psi = 0$ along a stationary wall)
  • Specifying the normal derivative of the stream function at inflow or outflow boundaries (e.g., $\frac{\partial \psi}{\partial n} = 0$ for a free stream boundary)
• The choice of boundary conditions depends on the specific flow problem and the available information about the flow

Applications of stream function

Visualization of flow patterns

• Contours of constant stream function values provide a visual representation of the flow patterns
• In two-dimensional flow, streamlines coincide with the contours of the stream function
• This visualization technique helps in understanding the qualitative behavior of the flow, such as the presence of recirculation zones or stagnation points

Calculation of mass flow rate

• The difference in stream function values between two points can be used to calculate the mass flow rate across a line connecting those points
• In two-dimensional flow, the mass flow rate per unit depth $\dot{m}'$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
  • $\dot{m}' = \rho (\psi_2 - \psi_1)$
  • where $\rho$ is the fluid density, and $\psi_1$ and $\psi_2$ are the stream function values at the two points
• This property is useful in determining the flow rate through channels, pipes, or around objects

Determination of stagnation points

• Stagnation points are locations in the flow where the velocity is zero
• In two-dimensional flow, stagnation points can be identified as the points where the contours of the stream function intersect orthogonally
• The stream function values at stagnation points are local extrema (maxima, minima, or saddle points)
• Identifying stagnation points is important in understanding the flow behavior and the forces acting on objects immersed in the flow

Relationship between stream function and other flow properties

Stream function and vorticity

• Vorticity $\omega$ is a measure of the local rotation in the fluid and is defined as the curl of the velocity vector: $\omega = \nabla \times \vec{V}$
• In two-dimensional flow, the vorticity is related to the stream function by:
  • $\omega = -\nabla^2 \psi$
• This relationship shows that the vorticity can be determined from the stream function, and vice versa, if the appropriate boundary conditions are known

Stream function and circulation

• Circulation $\Gamma$ is a scalar quantity that represents the line integral of the velocity vector along a closed curve
• In two-dimensional flow, the circulation around a closed curve $C$ is related to the stream function by:
  • $\Gamma = \oint_C \vec{V} \cdot d\vec{l} = \psi_B - \psi_A$
  • where $\psi_A$ and $\psi_B$ are the stream function values at any two points $A$ and $B$ on the curve $C$
• This relationship is a consequence of the constant value of the stream function along streamlines and is useful in analyzing the lift generated by airfoils using the Kutta-Joukowski theorem

Advantages and limitations of stream function approach

Simplification of flow analysis

• The stream function approach simplifies the analysis of incompressible and irrotational flows by reducing the number of dependent variables
• Instead of solving for the velocity components directly, the stream function can be determined, and the velocity components can be derived from it
• This simplification is particularly useful in two-dimensional flows, where the stream function is a scalar quantity

Limitations in compressible and unsteady flows

• The stream function approach is limited to incompressible flows, where the density is constant
• In compressible flows, where the density varies, the continuity equation cannot be satisfied by the stream function alone, and additional equations (e.g., the energy equation) must be considered
• The stream function approach is also limited to steady flows, where the flow properties do not change with time
• In unsteady flows, the stream function must be modified to account for the time-dependence of the flow, leading to more complex formulations (e.g., the unsteady stream function or the velocity potential)
• Despite these limitations, the stream function remains a valuable tool for analyzing and understanding a wide range of fluid flow problems, particularly in incompressible and irrotational flows