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Velocity potential

In fluid dynamics, the velocity potential is a scalar function \phi that describes the velocity field \mathbf{V} of an irrotational flow through the relation \mathbf{V} = \nabla \phi, ensuring zero vorticity \nabla \times \mathbf{V} = 0.^[1] This concept applies to inviscid, incompressible fluids, simplifying the analysis by reducing the vector velocity description to a single scalar potential that satisfies Laplace's equation \nabla^2 \phi = 0 under the continuity equation for incompressible flow.^[2] The velocity components are then given by u = \partial \phi / \partial x, v = \partial \phi / \partial y, and w = \partial \phi / \partial z in three dimensions.^[3] For irrotational flows, the velocity potential enables the superposition of elementary solutions—such as uniform flows, sources, sinks, and vortices—to model complex configurations like flow around airfoils or lifting bodies, forming the basis of potential flow theory in aerodynamics.^[3] In two-dimensional cases, it pairs with the stream function \psi to form a complex potential w(z) = \phi + i\psi, where equipotential lines (constant \phi) are orthogonal to streamlines (constant \psi), facilitating graphical and analytical solutions via the Cauchy-Riemann equations.^[1] This framework also integrates with Bernoulli's equation along streamlines to relate velocity, pressure, and other thermodynamic properties, though it assumes idealized conditions without viscosity or turbulence.^[2] Key applications include external aerodynamics, where potential flow approximations predict lift on slender bodies, acoustics for modeling sound wave propagation, and groundwater flow modeling, extending the concept beyond gases to porous media under Darcy's law analogies.^[3]^[4]^[5] Limitations arise in real fluids due to boundary layers and separation, prompting corrections like the Kutta-Joukowski condition for circulation in lifting flows.^[2] Overall, the velocity potential remains a cornerstone for theoretical fluid mechanics, bridging mathematical elegance with practical engineering insights.^[1]

Definition and Basics

Definition in Fluid Dynamics

In fluid dynamics, the velocity potential is defined as a scalar function \phi(\mathbf{x}, t), where \mathbf{x} = (x, y, z) denotes the position vector and t is time, such that the velocity field \mathbf{v} of the fluid is given by the gradient of this function: \mathbf{v} = \nabla \phi.^[6] This formulation expresses the three-dimensional velocity components as partial derivatives: u = \partial \phi / \partial x, v = \partial \phi / \partial y, and w = \partial \phi / \partial z.^[6] The velocity potential thus provides a compact way to describe the flow velocity using a single scalar field rather than a vector field. The units of the velocity potential \phi are those of velocity multiplied by length, which in SI units is square meters per second (m²/s).^[7] For instance, in the expression for uniform flow, \phi = U x where U is the uniform velocity (m/s) and x is the position coordinate (m), the product yields m²/s.^[7] This concept was introduced by the mathematician Joseph-Louis Lagrange in the late 18th century, specifically in 1781, as a means to simplify the Euler equations of motion for incompressible fluids by reducing them to a more manageable form involving potentials.^[8] Unlike the stream function \psi, which is a scalar used primarily for two-dimensional incompressible flows and defined such that the velocity components satisfy \mathbf{v} = \nabla \times (\psi \mathbf{k}) (analogous to a vector potential in the plane), the velocity potential directly yields the velocity via its gradient and is suited to irrotational flows.^[9] Such a representation is valid only for irrotational flows, where the vorticity \nabla \times \mathbf{v} = 0.^[6]

Prerequisites for Irrotational Flow

Irrotational flow is characterized by the condition that the curl of the velocity field vanishes everywhere, mathematically expressed as \nabla \times \mathbf{v} = 0, which implies zero vorticity (\boldsymbol{\omega} = 0) throughout the fluid domain.^[3] This absence of vorticity means that fluid elements do not rotate about their own axes, allowing the flow to be described without rotational components.^[2] The application of velocity potential requires specific assumptions about the fluid and flow conditions. Primarily, the fluid is assumed to be inviscid, meaning zero viscosity and no shear stresses, which simplifies the governing equations by neglecting frictional effects.^[2] Additionally, the flow is often treated as incompressible with constant density, though extensions to compressible flows are possible under certain subsonic conditions; the flow can be steady or unsteady, as long as the irrotational condition holds.^[10] These assumptions are particularly valid for external flows around bodies where viscous effects are minimal away from boundaries.^[3] From vector calculus, an irrotational vector field admits a scalar potential in simply connected domains, where the velocity field can be expressed as the gradient of a scalar function, \mathbf{v} = \nabla \phi, provided the field is continuously differentiable.^[11] This theorem ensures the existence and uniqueness (up to a constant) of the velocity potential \phi for such flows, as the domain's topology prevents issues like non-zero circulation around closed loops.^[10] Examples of flows satisfying these prerequisites include uniform flow, where the velocity is constant, and source or sink flows, which model radial expansion or contraction without rotation.^[2] In contrast, the irrotational assumption does not hold for turbulent flows or regions with concentrated vorticity, such as the cores of real vortices.^[3]

Mathematical Formulation

Relation to Velocity Field

In irrotational fluid flows, the velocity field \mathbf{v} is the gradient of the scalar velocity potential \phi, such that \mathbf{v} = \nabla \phi.^[12] This relation holds under the prerequisite that the flow is irrotational, meaning the curl of the velocity vanishes (\nabla \times \mathbf{v} = 0). In Cartesian coordinates (x, y, z), the velocity components are given by:

v_x = \frac{\partial \phi}{\partial x}, \quad v_y = \frac{\partial \phi}{\partial y}, \quad v_z = \frac{\partial \phi}{\partial z}.

^[13] In cylindrical coordinates (r, \theta, z), where r is the radial distance, \theta the azimuthal angle, and z the axial coordinate, the components are:

v_r = \frac{\partial \phi}{\partial r}, \quad v_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta}, \quad v_z = \frac{\partial \phi}{\partial z}.

^[13] In spherical coordinates (r, \theta, \varphi), where r is the radial distance, \theta the polar angle, and \varphi the azimuthal angle, the components are:

v_r = \frac{\partial \phi}{\partial r}, \quad v_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta}, \quad v_\varphi = \frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \varphi}.

^[13] For unsteady flows, the velocity potential \phi depends on both position and time, \phi(\mathbf{x}, t), so the velocity components remain spatial gradients, but the time derivative \partial \phi / \partial t contributes to pressure calculations via the unsteady Bernoulli equation.^[12] The velocity potential is unique up to an additive constant, as adding a constant to \phi does not alter the velocity field \mathbf{v}.^[14]

Derivation of Laplace's Equation

In incompressible fluid dynamics, the continuity equation states that the divergence of the velocity field \mathbf{v} is zero, \nabla \cdot \mathbf{v} = 0, reflecting the conservation of mass for a fluid of constant density.^[2] For irrotational flows, the velocity field can be expressed as the gradient of a scalar velocity potential \phi, such that \mathbf{v} = \nabla \phi.^[9] Substituting this into the continuity equation yields \nabla \cdot (\nabla \phi) = \nabla^2 \phi = 0, which is Laplace's equation governing the velocity potential.^[2] This derivation assumes incompressible flow, where density variations are negligible, leading to the elliptic partial differential equation \nabla^2 \phi = 0.^[3] In compressible flows, the continuity equation incorporates density changes, resulting in more complex forms such as the Prandtl-Glauert equation for the velocity potential under linearized assumptions, \left(1 - M_\infty^2\right) \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0, where M_\infty is the freestream Mach number; however, the incompressible case remains the primary focus for many potential flow analyses due to its analytical tractability.^[15] To solve Laplace's equation, appropriate boundary conditions must be specified. Dirichlet conditions prescribe the value of \phi directly on the boundary, while Neumann conditions specify the normal derivative \frac{\partial \phi}{\partial n}, which corresponds to the normal component of the velocity on impermeable surfaces where it is typically zero.^[16] These conditions ensure the solution satisfies the physical constraints of the flow domain.^[17] Common solution methods for Laplace's equation in potential flow include separation of variables, which decomposes the problem into simpler ordinary differential equations in separable coordinates like Cartesian, cylindrical, or spherical systems.^[18] For two-dimensional problems, complex potentials offer an elegant approach by representing \phi as the real part of an analytic function of the complex variable z = x + iy, leveraging conformal mapping properties.^[19]

Applications in Fluid Dynamics

Potential Flow Theory

Potential flow theory provides a framework for analyzing irrotational and inviscid fluid motions, where the velocity field is the gradient of a scalar velocity potential \phi that satisfies Laplace's equation \nabla^2 \phi = 0, as derived from the continuity and irrotationality conditions. This approach simplifies the description of incompressible flows by reducing the vector velocity field to a single scalar function, enabling the construction of complex flow patterns through the linear superposition of simpler, elementary solutions. Each elementary flow individually satisfies Laplace's equation, allowing their potentials to be added to model realistic configurations without violating the governing principles.^[20]^[7] Key elementary solutions include uniform flow, source/sink flows, and doublets. For uniform flow in two dimensions, the velocity potential is given by \phi = U x, where U is the constant speed in the x-direction, yielding a constant velocity \mathbf{u} = U \hat{i}. A two-dimensional source (or sink, with negative strength) has potential \phi = \frac{m}{2\pi} \ln r, where m is the source strength representing the volume flow rate per unit depth, and r = \sqrt{x^2 + y^2} is the radial distance from the source; the radial velocity is u_r = \frac{m}{2\pi r}. In three dimensions, a doublet—a limiting case of a source-sink pair—has potential \phi = \frac{\mu \cos \theta}{r}, where \mu is the doublet strength, \theta is the polar angle, and r is the radial distance, producing a flow pattern akin to that around a slender body.^[20]^[7]^[12] In two-dimensional flows, the complex potential w(z) = \phi + i \psi combines the velocity potential \phi with the stream function \psi, where z = x + i y is the complex coordinate. Analytic functions of z satisfy the Cauchy-Riemann conditions, ensuring both \phi and \psi are harmonic and thus solutions to Laplace's equation; this formulation facilitates conformal mapping to transform simple geometries into complex ones while preserving flow properties. For instance, the complex potential for uniform flow is w(z) = U z, for a source it is w(z) = \frac{m}{2\pi} \ln z, and for a doublet it is w(z) = -\frac{\mu}{2\pi z}.^[20]^[7] A classic example is the irrotational flow around a circular cylinder, obtained by superposing uniform flow and a doublet aligned with the oncoming flow. The resulting complex potential is w(z) = U \left( z + \frac{R^2}{z} \right), where R is the cylinder radius, yielding the velocity potential in polar coordinates as \phi = U \left( r + \frac{R^2}{r} \right) \cos \theta. The velocity field components are u_r = U \left( 1 - \frac{R^2}{r^2} \right) \cos \theta and u_\theta = -U \left( 1 + \frac{R^2}{r^2} \right) \sin \theta, with the boundary condition of no normal flow satisfied on the cylinder surface (r = R). This superposition models the external flow without penetration, providing the velocity distribution for further analysis.^[20]^[7]^[12]

Bernoulli's Equation Integration

In potential flow theory, the velocity potential \phi enables the integration of the Euler equations into the unsteady Bernoulli equation, which expresses the conservation of energy for irrotational, inviscid flows. This equation takes the form

\frac{\partial \phi}{\partial t} + \frac{1}{2} |\nabla \phi|^2 + \frac{P}{\rho} + gz = F(t),

where \phi is the velocity potential, P is the pressure, \rho is the fluid density, g is the gravitational acceleration, z is the elevation, and F(t) is an arbitrary function of time determined by boundary conditions.^[21] The term \frac{\partial \phi}{\partial t} accounts for temporal variations in the flow, while \nabla \phi represents the velocity field \mathbf{v}. This formulation arises from integrating the momentum equation along a streamline, leveraging the irrotational condition \nabla \times \mathbf{v} = 0 to simplify the convective acceleration.^[2] For steady flows, where \frac{\partial \phi}{\partial t} = 0, the equation simplifies to

\frac{1}{2} v^2 + \frac{P}{\rho} + gz = \text{constant},

with the constant uniform throughout the flow domain due to the irrotational nature, rather than merely along individual streamlines.^[22] Here, the kinetic energy per unit mass is directly computed as \frac{1}{2} v^2 = \frac{1}{2} |\nabla \phi|^2, allowing pressure P to be determined once \phi is solved from Laplace's equation. This integration facilitates efficient pressure distribution calculations in applications like aerodynamics, where solving for \phi first yields both velocity and pressure fields.^[2] The validity of these equations is restricted to irrotational and inviscid flows; they fail in the presence of vorticity or viscosity, as the potential representation breaks down and energy dissipation occurs.^[21] Additionally, singularities in the potential, such as at points of infinite velocity, invalidate the assumptions, though the formulation holds at stagnation points where \nabla \phi = 0 and v = 0, resulting in stagnation pressure P_s = P_\infty + \frac{1}{2} \rho v_\infty^2 for steady flow past a body.^[2] For instance, in uniform flow approaching a stagnation point on a surface, the maximum pressure reflects the full conversion of freestream kinetic energy to pressure.

Applications in Other Fields

Acoustics

In acoustics, the velocity potential is employed under the acoustic approximation, which assumes small perturbations in pressure, density, and velocity relative to the ambient fluid state, such that |p'| \ll p_0, |v'| \ll c, and |\rho'| \ll \rho_0, where primes denote perturbations, p_0 and \rho_0 are ambient pressure and density, and c is the speed of sound.^[4] This linearization of the Euler and continuity equations leads to an irrotational flow where the particle velocity is the gradient of the scalar velocity potential \mathbf{v} = \nabla \phi, with the acoustic pressure given by p = -\rho_0 \frac{\partial \phi}{\partial t}.^[23] Unlike the steady or unsteady hydrodynamic potential for incompressible flows, the acoustic version accounts for compressibility through finite wave propagation speeds, enabling time-dependent wave phenomena.^[23] Combining the linearized momentum equation (neglecting viscous terms) with the continuity equation yields the wave equation for the velocity potential:

\nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = 0,

where the Laplacian \nabla^2 \phi arises from the divergence of the velocity, and the time derivative term incorporates compressibility via the speed of sound c = \sqrt{\gamma p_0 / \rho_0} for an ideal gas.^[4] For time-harmonic fields assuming e^{-i\omega t} dependence, this reduces to the Helmholtz equation \nabla^2 \phi + k^2 \phi = 0, with wavenumber k = \omega / c.^[23] A fundamental solution is the plane wave, expressed as \phi = A \cos(\mathbf{k} \cdot \mathbf{x} - \omega t), where A is the amplitude, \mathbf{k} is the wave vector with |\mathbf{k}| = k, and \omega = c k ensures dispersionless propagation.^[4] The corresponding pressure is p = \rho_0 \omega A \sin(\mathbf{k} \cdot \mathbf{x} - \omega t), yielding the characteristic acoustic impedance Z = p / v = \rho_0 c along the propagation direction for progressive waves.^[23] This formulation finds applications in modeling sound radiation from compact sources, such as monopoles representing volume velocity sources, where the far-field potential decays as $1/r and pressure as $1/r, facilitating power calculations via the radiation impedance Z_r = p / U (with U as source volume velocity).^[4] Impedance boundary conditions, \frac{\partial \phi}{\partial n} = \frac{i k \rho_0 c}{Z} \phi, are applied to surfaces with specified admittance (noting conventions for time dependence and normal direction), enabling analysis of reflections and absorption in enclosures.^[24] Historically, Lord Rayleigh pioneered this approach in the late 19th century, deriving the velocity potential wave equation for aerial vibrations in his seminal two-volume work The Theory of Sound (1877–1878), which laid foundational principles for modern acoustic theory.^[25]

Groundwater Hydrology

In groundwater hydrology, the velocity potential adapts concepts from fluid dynamics to model slow, viscous flow through porous media, analogous to Darcy's law, which states that the specific discharge (Darcy velocity) \mathbf{q} is proportional to the negative gradient of the hydraulic head h: \mathbf{q} = -K \nabla h, where K is the hydraulic conductivity.^[26] This law, empirically derived from experiments on sand filters, assumes laminar flow and linear proportionality between flow rate and head difference.^[26] The velocity potential \phi is defined such that \mathbf{q} = -\nabla \phi, with \phi = K h (or sometimes \phi = -K h depending on sign convention; here \phi = K h is used), transforming the vector field of specific discharge into a scalar potential that satisfies irrotational flow conditions when the medium is isotropic. For 2D flow in confined aquifers, a discharge potential \Psi = T h (with transmissivity T = K b, b aquifer thickness) is sometimes used, where the discharge per unit width \mathbf{q}' = -\nabla \Psi.^[27]^[28] This formulation enables the use of potential theory to analyze subsurface water movement, where \phi has units of velocity times length (e.g., m²/s) and represents the energy potential scaled by permeability.^[28] For steady-state, incompressible flow in confined aquifers or under simplifying assumptions for unconfined ones, the continuity equation combined with Darcy's law yields Laplace's equation for the velocity potential: \nabla^2 \phi = 0.^[26] This elliptic partial differential equation governs the potential in homogeneous, isotropic media without sources or sinks, allowing solutions via analytical methods or numerical flow nets. In unconfined aquifers, the Dupuit-Forchheimer assumptions approximate the flow by neglecting vertical velocity components and assuming horizontal flow lines parallel to the base, which linearizes the free-surface boundary and permits application of Laplace's equation to \phi.^[29] These assumptions, introduced by Dupuit in 1863 for seepage problems, hold best for shallow slopes and high-permeability media but introduce errors near the water table.^[29] Applications of velocity potential include modeling flow in confined and unconfined aquifers, where equipotential lines and flow paths form orthogonal networks for seepage analysis. For instance, in seepage under dams or sheet piles, flow nets constructed from \phi contours quantify discharge rates and identify zones of uplift pressure.^[28] In pumping well scenarios, steady radial flow to a well in a confined aquifer follows the Thiem equation, derived in the late 19th century: \phi = \frac{Q}{2\pi b} \ln r + C, where Q is the pumping rate, r is radial distance, b is the aquifer thickness, and C is a constant, enabling estimation of transmissivity T = K b from observed head drawdown.^[26]^[30] For transient conditions, the Theis solution extends the framework by solving the groundwater flow equation (a diffusion equation) to predict time-dependent drawdown around wells, briefly incorporating potential-like terms under initial steady assumptions before transient storage effects dominate.^[26] The use of velocity potential in groundwater hydrology emerged in the 19th and early 20th centuries, building on Darcy's 1856 experiments and Dupuit's 1863 theoretical advancements for unconfined flow, with Thiem's 1906 work formalizing steady-state well hydraulics.^[30]^[29] These developments integrated potential theory into hydrogeology, influencing modern aquifer testing and seepage design. However, the approach assumes isotropic and homogeneous media, which rarely holds in fractured or layered aquifers, and neglects transient storage changes, limiting accuracy for time-varying flows or anisotropic conditions.^[26]^[28]

References

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An Internet Book on Fluid Dynamics. Velocity Potential. Irrotational flow is defined as a flow in which the vorticity, ω, is zero and since ω = ∇ × u. (Bdf1).
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When flow is irrotational it reduces nicely using the potential function in place of the velocity vector. The potential function can be substituted into ...
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This has units of m 2 / s in SI units. ... It would be a good exercise to generate plots of the above equations for velocity potential and streamfunction.
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### Summary of Lagrange and Velocity Potential Introduction
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We conclude that, for two-dimensional, irrotational, incompressible flow, the velocity potential and the stream function both satisfy Laplace's equation.Missing: definition | Show results with:definition
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In lecture a detailed discussion is given in which the linearized velocity potential equation is obtained: 1 − M2. ∞ ∂. 2φ. ∂x2. +. ∂2φ. ∂y2. = 0. (6.1). 6.2 ...<|control11|><|separator|>
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Uniform flow, Sources, Sinks, Doublets. Reading: Anderson 3.9 – 3.12. Uniform Flow. Definition. A uniform flow consists of a velocity field where. V. = uı+ vˆ ...
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None
### Summary of Unsteady Bernoulli Equation Derivation
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It works especially for simple cases such as impulsively started confined flows where streamlines have the same shape at each instant and we are interested in ...
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In other publications, equipotentials may be labelled with values of dimension [L2T−1], and in this case represent the velocity potential (KH + constant).<|separator|>
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