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Stream Function

Stream Function

In two-dimensional incompressible fluid flow, the continuity equation is essential for ensuring mass conservation, meaning that any change in fluid entering or exiting a region is balanced by a corresponding change elsewhere. For incompressible flow, where density remains constant, this requirement simplifies to the condition that the divergence of the velocity field must be zero. Mathematically, this is expressed as,

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:

Bernoulli's Equation for Flow Normal to a Streamline

Bernoulli's Equation for Flow Normal to a Streamline

Bernoulli's equation for flow normal to a streamline explains how pressure varies across curved streamlines due to the outward centrifugal forces induced by the fluid's curvature. The pressure is higher on the inner side of the curve, near the center of curvature, and decreases outward to balance these centrifugal forces.
The pressure difference depends on the fluid's velocity and radius of curvature. The pressure variation is minimal in flows with nearly straight streamlines.

Velocity Potential

Velocity Potential

In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:

Couette Flow

Couette Flow

Couette flow represents the flow of fluid between two parallel plates, with one plate fixed and the other moving with a constant velocity. This configuration allows for a simplified analysis using the Navier-Stokes equations, which govern fluid motion under conditions of viscosity and incompressibility. For Couette flow, the assumptions include a steady, laminar, incompressible flow with a zero-pressure gradient in the flow direction. This flow type is beneficial for understanding shear-driven...

Steady Flow of a Fluid Stream

Steady Flow of a Fluid Stream

Consider a control volume, such as a pipe with solid boundaries, through which fluid flows and changes direction due to the impulse exerted by the resulting force from the pipe walls. In steady flow, the mass of fluid entering the control volume at a given time, t, with velocity v1, is equal to the mass leaving after infinitesimal time dt, with velocity v2.
During this process, the momentum of the fluid within the control volume remains constant over the time interval dt. By applying the...

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Related Experiment Video

Updated: Oct 19, 2025

Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel

Uncoupling Coriolis Force and Rotating Buoyancy Effects on Full-Field Heat Transfer Properties of a Rotating Channel

Published on: October 5, 2018

Buoyant Convection Computed in a Vorticity, Stream-Function Formulation.

Ronald G Rehm¹, Howard R Baum¹, P Darcy Barnett¹

¹National Bureau of Standards, Washington, DC 20234.

Journal of Research of the National Bureau of Standards (1977)

|September 27, 2021

Summary

This summary is machine-generated.

This study presents a novel computational model for buoyant convection in enclosures, neglecting viscosity and thermal conductivity. The model accurately simulates fluid flow driven by heat sources, validated against primitive equation results.

Keywords:

Lanezos smoothing buoyant convection finite difference computations fire-enclosure fluid flow partial differential equations stream function vorticity

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Published on: July 19, 2016

Area of Science:

Fluid Dynamics
Computational Physics
Heat Transfer

Background:

Buoyant convection is crucial in various natural and engineered systems.
Previous models often include viscous and thermal conductivity effects, limiting applicability.
Thermally expandable fluid models are needed for large density variations without acoustic waves.

Purpose of the Study:

To formulate and solve model equations for large-scale buoyant convection.
To develop a unique vorticity-stream function formulation for this problem.
To validate the computational algorithm against established methods.

Main Methods:

Formulation of model equations using vorticity and stream function.
Neglecting viscous and thermal conductivity effects.
Employing a thermally expandable fluid model to filter acoustic waves.
Developing and presenting a novel algorithm for the vorticity-stream function formulation.
Utilizing Lanczos smoothing for computational results.

Main Results:

The developed algorithm successfully solves the buoyant convection equations.
Computational results show excellent agreement with solutions from primitive equations.
The study demonstrates the effectiveness of Lanczos smoothing in computations.

Conclusions:

The novel vorticity-stream function model provides an accurate and efficient method for simulating buoyant convection.
The approach is suitable for scenarios with large density variations.
The presented algorithm and smoothing technique enhance computational fluid dynamics capabilities.