The application of variational principles to fluid mechanics has a rich history spanning over 150 years, beginning with Helmholtz’s vortex theorems (1858) and Kelvin’s circulation theorem (1869). Classical approaches typically employed Hamilton’s principle, extremising action integrals over time to derive the equations of motion through calculus of variations (Holm, Marsden & Ratiu Reference Holm, Marsden and Ratiu1998; Truesdell & Noll Reference Truesdell and Noll2004; Marchioro & Pulvirenti Reference Marchioro and Pulvirenti2012). These time-integral formulations, while mathematically elegant, vary entire space–time paths and thus provide limited insight into the instantaneous mechanics of constraint enforcement in incompressible flows (Holm et al. Reference Holm, Marsden and Ratiu1998; Truesdell & Noll Reference Truesdell and Noll2004; Marchioro & Pulvirenti Reference Marchioro and Pulvirenti2012).
A fundamentally different variational approach emerged from Gauss’s principle of least constraint (Gauss Reference Gauss1829), originally developed for constrained mechanical systems. Unlike Hamilton’s principle which varies paths over time, Gauss’s principle operates at a fixed instant, minimising a quadratic form in accelerations subject to acceleration-level constraints. This instantaneous character makes it particularly suited to understanding how pressure maintains incompressibility moment by moment. Despite its potential, the principle remained relatively uncommon in classical fluid mechanics for over a century, even though it is widely used in constrained molecular dynamics (e.g. Nosé Reference Nosé1984; Hoover Reference Hoover1985; Evans & Morris Reference Evans and Morris1990; Thijssen Reference Thijssen2007). Recent works (Gonzalez & Taha Reference Gonzalez and Taha2022; Taha, Gonzalez & Shorbagy Reference Taha, Gonzalez and Shorbagy2023; Taha & Shorbagy Reference Taha and Shorbagy2025; Peters & Ormiston Reference Peters and Ormiston2025) have brought back a variational lens on fluid flows, particularly revisiting the Gauss principle in the context of the airfoil lift problem.
Despite the renewed interest, there remains confusion about what a Gauss/Appell least-constraint statement actually determines in fluids, and how it can be utilised. We provide a compact variational interpretation of the pressure Poisson equation, clarify how to embed physical inputs via an ‘impressed’ subspace and give a corollary that places the projection method in numerical hydrodynamics (Chorin Reference Chorin1968) under Gauss/Appell.
Gonzalez & Taha (Reference Gonzalez and Taha2022) minimise a steady reduced functional on a kinematically admissible manifold (e.g. a potential-flow family parameterised by circulation), a model-reduction approach but not the instantaneous Gauss principle. Taha et al. (Reference Taha, Gonzalez and Shorbagy2023) derive a general ‘minimum pressure-gradient’ interpretation directly from Gauss’s principle (for arbitrary viscous stresses), showing that Navier–Stokes arises as the first-order optimality condition when the variation is taken with respect to the local acceleration. Steady/inviscid specialisations of such costs have also been explored as selection mechanisms layered on top of Euler (e.g. Taha & Shorbagy Reference Taha and Shorbagy2025). In particular, the fixed-time Gauss/Appell projection step considered here does not, by itself, resolve classical indeterminacies of steady exterior potential flow, because those indeterminacies live in the choice of steady velocity state rather than in the instantaneous constrained acceleration correction. However, each candidate steady state has an associated reaction pressure whose magnitude depends on the circulation; this was exploited as a basis for secondary minimisation, as explored in the variational theory of lift (Gonzalez & Taha Reference Gonzalez and Taha2022; Taha et al. Reference Taha, Gonzalez and Shorbagy2023; Taha & Shorbagy Reference Taha and Shorbagy2025).
This work clarifies these viewpoints by showing precisely where Gauss/Appell ends (fixed-time projection) and where additional modelling/selection begins. The Gauss principle is a fixed-time variational statement: one freezes the velocity field and varies the material acceleration. Minimising a positive–definite ‘Appellian’ in the admissible accelerations yields first-order optimality conditions that are exactly the Leray–Hodge projection of the residual onto gradient fields; the Lagrange multiplier is a reaction pressure obtained from a Poisson–Neumann problem whose gradient removes the non-solenoidal part of the residual. In steady flow, this principle does not select among kinematically admissible steady states; it only determines the pressure compatible with the given velocity field. These points are well known from projection methods but are often obscured in recent discussions; we make the equivalence and its scope precise.
Further, we clarify that shifting a known scalar potential between the two terms in
$p=P_F+P_R$
is an algebraic invariance of the Gauss–Appell optimality conditions: the optimiser
$u_t^\star$
and the total pressure
$p$
are unchanged. This invariance does not imply physical non-uniqueness of the reaction (constraint) pressure. Building on Peters & Ormiston (Reference Peters and Ormiston2025), we show how an energy (Dirichlet) orthogonality condition can be used as a gauge-fixing convention to pick a unique representative once a reference/impressed subspace is chosen, and connect this to the classical Leray projection. When
$P_F$
is reserved for truly impressed physics (e.g. conservative body-force potentials specified independently of incompressibility), the split is unique and
$P_R$
is the unique Lagrange multiplier enforcing the constraints (up to a constant). This is the well-known result that the pressure serves as the Lagrange multiplier for the incompressibility constraint (Gresho & Sani Reference Gresho and Sani1987; Sanders et al. Reference Sanders, DeVoria, Washuta, Elamin, Skenes and Berlinghieri2024), a fact established since the early work of Lagrange himself (Morrison Reference Morrison2020).
Unlike the principle of least action, Gauss/Appell is a least-deviation principle in the accelerations at a given instant. In this sense, it is clarified that Gauss/Appell provides a unifying variational interpretation of the pressure Poisson solve – useful analytically and practically, particularly when one wishes to embed physical inputs (e.g. circulation or far-field content) through an impressed subspace while keeping the constraints exact. We present rigorous derivations and provide insight into the physical implications of the Gauss principle.
1. Setting, notations
Let
$\varOmega$
be the exterior domain of a smooth body, and
$\partial \varOmega$
be the boundaries. The density
$\rho$
is constant;
$u(\boldsymbol{\cdot },t)$
is the velocity field. Gauss’ principle is applied at a fixed time. One freezes the state
$u(\boldsymbol{\cdot },t)$
and chooses an admissible instantaneous update in acceleration. For fluids, the material acceleration is
$ a \triangleq \partial _t u+(u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u.$
We assume throughout that
$u(\boldsymbol{\cdot },t)$
is sufficiently smooth such that the quadratic functional below is finite. At a fixed time, the convective part
$(u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u$
is a known function of space (because
$u(\boldsymbol{\cdot },t)$
is frozen). Hence a perturbation of acceleration comes entirely from perturbing
$u_t$
:
$\delta a=\delta u_t$
. This is the fluid analogue of Gauss in constrained multibody dynamics (Udwadia & Kalaba Reference Udwadia and Kalaba1992), where one minimises a positive–definite quadratic form in accelerations at a fixed configuration and velocity, subject to acceleration-level constraints; here the Eulerian fixed-time data are
$(u,\boldsymbol{\nabla }\!\boldsymbol{\cdot }u, u\!\boldsymbol{\cdot }n)$
. Therefore, the variational variable is
$u_t(\boldsymbol{\cdot },t)$
, not a time-slab history, and not a configuration displacement as in stationary action.
For an airfoil, with far-field velocity
$U_\infty$
and slip wall
\begin{align} \boldsymbol{\nabla }\!\boldsymbol{\cdot }u = 0\quad \text{in } \quad \varOmega ,\quad u\!\boldsymbol{\cdot }n = 0\quad \text{on } \quad \partial \varOmega ,\quad u\to U_\infty \quad \text{ as } \quad |x|\to \infty . \end{align}
At the fixed instant
$t$
, these are given. Differentiating the two kinematic constraints in time gives the acceleration-level constraints on
$u_t$
\begin{equation} \boldsymbol{\nabla }\!\boldsymbol{\cdot }u_t = 0\quad \text{in } \quad \varOmega ,\quad u_t\!\boldsymbol{\cdot }n = 0\quad \text{on } \quad \partial \varOmega . \end{equation}
Below, the Eulerian form of the derivation is detailed.
1.1. Impressed vs reaction pressure and the Appellian
Following the notation of Peters & Ormiston (Reference Peters and Ormiston2025), we write the total pressure
\begin{align} p \triangleq P_F+P_R, \end{align}
where
-
(i)
$P_F$
(impressed/prescribed) is a scalar potential associated with conservative impressed forcing that is specified independently of the incompressibility and wall constraints (e.g. a body-force potential); in unforced Euler one may take
$P_F\equiv 0$
; -
(ii)
$P_R$
(constraint) is the Lagrange multiplier that enforces the acceleration–level constraints completely and uniquely. Its gradient
$-\boldsymbol{\nabla }\!P_R$
is the constraint force per unit volume.
For a fixed velocity snapshot
$u(\boldsymbol{\cdot },t)$
and prescribed impressed potential
$P_F$
, the reaction pressure
$P_R$
is determined uniquely (up to an additive constant) by the Poisson–Neumann problem (2.7). In the linearised unforced setting emphasised in Peters & Ormiston (Reference Peters and Ormiston2025), taking the divergence yields
${\nabla} ^2 p=0$
, so only the sum
$p=P_F+P_R$
is constrained to be harmonic.
The stronger separation
${\nabla} ^2 P_F={\nabla} ^2 P_R=0$
(Eq. (17) of Peters & Ormiston (Reference Peters and Ormiston2025)) is therefore an additional harmonic-gauge convention within their formal decomposition. We stress that a true impressed force (e.g. electromagnetic
$\boldsymbol{\nabla }\varphi$
) is prescribed independently of the incompressibility constraint and need not satisfy
${\nabla} ^2\varphi =0$
. By contrast, constraint forces are precisely those that satisfy the Poisson/Laplace restrictions needed to enforce kinematic constraints. Therefore, if a pressure component is required to satisfy
${\nabla} ^2 P_F=0$
as a consequence of the continuity constraint, it is by definition not an impressed force in the sense of analytical mechanics; see also Taha (Reference Taha2026) for a discussion of this point. In the present work we do not adopt such restrictions on
$P_F$
; once any impressed conservative forcing is specified, the reaction pressure
$P_R$
enforcing (1.2) is uniquely determined (up to a constant) by (2.7).
Define the frozen field
\begin{equation} C \triangleq (u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u + \frac {1}{\rho }\,\boldsymbol{\nabla }\!P_F, \end{equation}
which is entirely known once
$u(\boldsymbol{\cdot },t)$
and
$P_F(\boldsymbol{\cdot },t)$
are specified. Gauss/Appell at the instant minimises the Appellian
The quadratic form in (1.5) uses the mass-weighted
$L^2$
inner product on accelerations, and yields the classical Leray–Hodge projection. In the function-space setting this choice is not unique: other metrics define different least-constraint projections. Weaker norms and lower regularity settings have been used in variational approaches to high-Reynolds-number flows; see e.g. Hoffman & Johnson (Reference Hoffman and Johnson2010), Hoffman, Jansson & Johnson (Reference Hoffman, Jansson and Johnson2016).
2. First-order stationarity
We write the constrained Lagrangian and derive the first-order Karush–Kuhn–Tucker (KKT) conditions. The Lagrangian with multipliers
$\lambda$
(volume) and
$\mu$
(boundary) is
\begin{align} \mathcal L[u_t,\lambda ,\mu ] \triangleq \frac 12\int _\varOmega \rho \,|u_t+C|^2\,{\rm d}V \;-\;\int _\varOmega \lambda \,(\boldsymbol{\nabla }\!\boldsymbol{\cdot }u_t)\,{\rm d}V \;-\;\int _{\partial \varOmega } \mu \, (u_t\!\boldsymbol{\cdot }n)\,{\rm d}S. \end{align}
Variations yield
\begin{align} \delta _{\lambda }:\ \boldsymbol{\nabla }\!\boldsymbol{\cdot }u_t=0,\quad \delta _{\mu }:\ u_t\!\boldsymbol{\cdot }n=0, \end{align}
and
\begin{align} \delta _{u_t}\mathcal L &=\int _\varOmega \big (\rho (u_t+C)+\boldsymbol{\nabla }\lambda \big )\boldsymbol{\cdot }\delta u_t\,{\rm d}V \;+\;\int _{\partial \varOmega }(\lambda -\mu )\,\delta u_t\!\boldsymbol{\cdot }n\,{\rm d}S. \end{align}
Because
$\delta u_t$
is arbitrary in the interior and its normal component is arbitrary on
$\partial \varOmega$
(before enforcing the constraint), stationarity gives
\begin{equation} \rho (u_t+C)+\boldsymbol{\nabla }\lambda =0\quad \text{in } \quad \varOmega ,\quad \mu =\lambda \quad \text{on } \quad \partial \varOmega , \end{equation}
together with (1.2). This derivation follows the same constrained-optimisation/projection-step logic as Peters & Ormiston (Reference Peters and Ormiston2025), while emphasising that in the present work
$P_F$
denotes only prescribed conservative impressed forcing (if any) and the reaction pressure
$P_R$
carries all constraint-enforcing pressure content.
Now, define
$P_R \triangleq \lambda$
. From (2.4),
$u_t=-(C+\boldsymbol{\nabla }\!P_R/\rho )$
; impose (1.2)
\begin{align} \boldsymbol{\nabla }\!\boldsymbol{\cdot }u_t=0\ \Rightarrow \ {\nabla} ^2 P_R = -\,\rho \,\boldsymbol{\nabla }\!\boldsymbol{\cdot }C\quad \text{in } \quad \varOmega , \end{align}
and
\begin{align} u_t\!\boldsymbol{\cdot }n=0\ \Rightarrow \ \partial _n P_R = -\,\rho \,C\!\boldsymbol{\cdot }n\quad \text{on } \quad \partial \varOmega . \end{align}
Thus the first step is the scalar Poisson–Neumann boundary value problem
\begin{equation} \fbox{$ {\nabla} ^2 P_R = -\,\rho \,\boldsymbol{\nabla }\!\boldsymbol{\cdot }C \quad \text{ in } \quad \varOmega ,\quad \partial _n P_R = -\,\rho \,C\!\boldsymbol{\cdot }n \quad \text{ on } \quad \partial \varOmega $}. \end{equation}
Compatibility holds identically:
$ \int _\varOmega {\nabla} ^2 P_R\,{\rm d}V=\int _{\partial \varOmega }\partial _n P_R\,{\rm d}S$
because
$ \int _\varOmega \boldsymbol{\nabla }\!\boldsymbol{\cdot }C\,{\rm d}V=\int _{\partial \varOmega } C\!\boldsymbol{\cdot }n\,{\rm d}S.$
With (2.7) solved, we have
\begin{align} u_t^\star =-(C+\boldsymbol{\nabla }\!P_R/\rho ),\quad p=P_F+P_R. \end{align}
Since
$a=u_t+(u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u$
, (2.4) is equivalent to
\begin{align} \rho a =-\boldsymbol{\nabla }\!P \end{align}
i.e. Euler at the instant.
Once optimised, the minimised Appellian is purely ‘projection effort’
\begin{align} \mathcal A_\star =\frac 12\int _\varOmega \frac {1}{\rho }\,|\boldsymbol{\nabla }\!P_R|^2\,{\rm d}V. \end{align}
This reduced functional is the instantaneous
$L^2$
norm of the reaction-pressure gradient (the ‘minimum pressure-gradient’ quantity emphasised by Taha et al. (Reference Taha, Gonzalez and Shorbagy2023). This is precisely the standard projection method logic in continuous form: solve a Poisson–Neumann problem for pressure and project the residual onto gradients. It is notable that the Leray projector (see, for instance, Temam Reference Temam2024) maps any field to the divergence-free, wall-tangent subspace.
Writing
$ R\triangleq \rho \ a +\boldsymbol{\nabla }\!P_F=\rho (u_t+C),$
the KKT conditions
$\rho (u_t^\star +C)=-\boldsymbol{\nabla }\!P_R$
says
$R=-\boldsymbol{\nabla }\!P_R$
: the solenoidal, wall-tangent part of
$R$
vanishes. Thus
$P_R$
realises the Leray projection of
$R$
onto gradients, and (2.7) is the strong-form projector. This connection of the Leray projector to the Gauss–Appell principle appears not to have been noted in the literature.
3. Physical implications and insight
At a fixed instant, the Gauss–Appell step has a concrete physical meaning: it chooses the smallest admissible change to the material acceleration that restores the instantaneous momentum balance while preserving incompressibility and wall non-penetration. The only agency it has is a reaction field,
$-\boldsymbol{\nabla }\!P_R/\rho$
, whose role is to remove whatever divergence and normal flux the provisional residual carries. The pressure is a reaction that instantaneously enforces the kinematics; it does no work on divergence-free, wall-tangent motions. This point is correctly argued by Gonzalez & Taha (Reference Gonzalez and Taha2022).
3.1. Projection effort as a diagnostic
After elimination, the minimised Appellian
$\mathcal{A}^\ast$
becomes a quantitative measure of the projection effort, i.e. how hard the constraint must ‘push’ to keep the instantaneous update compatible with incompressibility and wall tangency. The minimised Appellian can be written as
\begin{align} \mathcal A_\star =\frac {1}{2\rho }\,\|\boldsymbol{\nabla }\!P_R\|_{L^2(\varOmega )}^2, \end{align}
and therefore provides a scalar measure of the magnitude of the constraint correction at a given instant. In particular, if the frozen residual
$C$
is already compatible with the acceleration constraints,
$\boldsymbol{\nabla }\!\boldsymbol{\cdot }C=0$
in
$\varOmega$
and
$C\!\boldsymbol{\cdot }n=0$
on
$\partial \varOmega$
, then (2.7) reduces to Laplace’s equation with homogeneous Neumann data and hence
$\boldsymbol{\nabla }\!P_R\equiv 0$
(up to an additive constant), so that
$\mathcal A_\star =0$
.
In computations, a sudden growth of
$\mathcal A_\star$
indicates that the provisional update violates divergence or wall-flux constraints and the projection is applying a large correction; this may occur, for example, due to inconsistent boundary treatment or under-resolution. More details on the projection operator in the context of general boundary conditions can be found in Maria Denaro (Reference Maria Denaro2003).
3.2. Steady flow
It is emphasised that in a steady flow, the Gauss principle of least constraint finds the gradient one must subtract from the ‘residual’
$\rho (u \boldsymbol{\cdot }\boldsymbol{\nabla }) u + \boldsymbol{\nabla }\!P_F$
so that the momentum equation holds. There is no update to
$u$
(the minimiser gives
$u_t^\ast =0$
) and no direct selection among multiple steady Euler fields that satisfy the kinematics.
However, since the minimised Appellian
$\mathcal{A}^\star = ({1}/{2\rho })\|\boldsymbol{\nabla }\!P_R\|^2_{L^2}$
depends on the frozen velocity state, different kinematically admissible steady fields (e.g. parameterised by circulation
$\varGamma$
) yield different values of this cost. This provides an indirect but fundamental connection between the Gauss cost and the selection of circulation, as explored in the variational theory of lift (Gonzalez & Taha Reference Gonzalez and Taha2022; Taha et al. Reference Taha, Gonzalez and Shorbagy2023).
Gonzalez & Taha (Reference Gonzalez and Taha2022) restrict the velocity field to a steady potential-flow family
$u(\boldsymbol{\cdot };\varGamma ) \triangleq u_0+\varGamma u_1$
whose members satisfy continuity and wall tangency, and then define
\begin{align} S(\varGamma ) \triangleq \frac 12\int _\varOmega \rho \,\big |\,u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla }u\,\big |^2\,{\rm d}V, \end{align}
seeking
$\varGamma ^\star =\arg \min _\varGamma S(\varGamma )$
. This is a reduced-order optimisation on a one-parameter manifold of kinematically admissible steady flows and appears to pick a
$\varGamma$
consistent with the Kutta condition in nearly sharp airfoils. However, it does not directly follow from the instantaneous Gauss/Appell principle; it presupposes steadiness and a particular parameterisation. Peters & Ormiston (Reference Peters and Ormiston2025) discuss cases where such criteria can select steady states that do not align with physical reality; see also Taha (Reference Taha2026) for a response to these claims.
In the full Gauss setting, the state
$u$
is frozen, the variable is
$u_t$
, and the outcome is the constraint projection
$P_R$
(§ 2), not the selection of
$\varGamma$
. In the steady case, for every
$\varGamma$
there exists a pressure that makes
$ (u(\boldsymbol{\cdot };\varGamma ),p(\boldsymbol{\cdot };\varGamma ))$
a steady Euler solution. Applying Gauss at the instant then yields
$u_t^\ast =0$
for every
$\varGamma$
. Therefore, the direct fixed-time application of Gauss’s principle does not, by itself, favour a particular value of
$\varGamma$
; it simply returns the reaction pressure associated with each already-specified steady velocity state. In the steady case, the fixed-time Gauss step adds no additional closure beyond computing the compatible pressure (any selection over steady states is additional modelling).
Taha et al. (Reference Taha, Gonzalez and Shorbagy2023) (see also Taha & Shorbagy (Reference Taha and Shorbagy2025)) propose the ‘principle of minimum pressure gradient’, and offer reduced-manifold selections that align with classical expectations from applying the Kutta condition on nearly sharp airfoils. An important distinction is required to interpret scope: Gauss/Appell provides the fixed-time projection; selection over steady states requires additional modelling choices.
In the unsteady case, since
$u_t$
(hence
$a$
) is varied with
$u$
held fixed, acceleration-level constraints are enforced, resulting in the reaction pressure via the Poisson–Neumann problem (2.7). This exactly recovers Euler at the instant and clarifies the roles of impressed vs constraint pressure. This also explains why the fixed-time least-constraint step does not, by itself, resolve the classical indeterminacy of steady exterior potential flow: the indeterminacy lives in the state manifold, not in the instantaneous constrained acceleration update. However, as noted above, the magnitude of the reaction pressure varies across the family of steady states, a fact that is exploited as a cost functional for a secondary selection principle (Gonzalez & Taha Reference Gonzalez and Taha2022; Taha et al. Reference Taha, Gonzalez and Shorbagy2023).
3.3. Representational invariance of the Gauss–Appell step
It is emphasised that, once the impressed physics is specified (including the case
$P_F\equiv 0$
when no body forces act), the reaction pressure
$P_R$
is uniquely determined (up to an additive constant) as the Lagrange multiplier enforcing incompressibility and wall impermeability. This is consistent with classical results (Gresho & Sani Reference Gresho and Sani1987; Sanders et al. Reference Sanders, DeVoria, Washuta, Elamin, Skenes and Berlinghieri2024).
With this physical uniqueness established, we note an algebraic property of the Gauss–Appell optimality conditions. Let
$\psi$
be any prescribed smooth scalar field (a reference pressure/gauge potential). Define a new split
$ \widetilde P_F \triangleq P_F + \psi ,\quad \widetilde P_R \triangleq P_R - \psi$
. Then
\begin{align} \widetilde C = (u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u + \frac {1}{\rho }\boldsymbol{\nabla }\!\widetilde P_F = C + \frac {1}{\rho }\boldsymbol{\nabla }\psi , \end{align}
and the first-order optimality condition becomes
\begin{align} \rho \,(u_t^\star +\widetilde C)+\boldsymbol{\nabla }\!\widetilde P_R = \rho \,(u_t^\star +C) + \boldsymbol{\nabla }\!P_R = 0, \end{align}
so the optimiser
$u_t^\star$
is invariant. Likewise
$p=\widetilde P_F+\widetilde P_R$
is unchanged. The formal decomposition
$p = P_F + P_R$
therefore admits a family of representatives for any given total pressure
$p$
: the equations alone do not fix the split.
Similarly, the Poisson equation changes to
${\nabla} ^2 \widetilde P_R = -\rho \,\boldsymbol{\nabla }\!\boldsymbol{\cdot }C - {\nabla} ^2\psi ,\ \partial _n\widetilde P_R = -\rho \,C\!\boldsymbol{\cdot }n - \partial _n\psi$
. Therefore
$\widetilde P_R = P_R - \psi + \text{const}$
, exactly as above. So the Poisson step leaves
$(u_t^\star ,p)$
invariant. This invariance is a purely algebraic property of the equations. It does not imply physical non-uniqueness of the constraint force, because the shifted quantity
$\boldsymbol{\nabla }\psi$
cannot qualify as a physical impressed force in the sense of analytical mechanics: a genuine impressed force must be prescribed independently of the constraint, and must persist if the constraint is removed.
If the incompressibility constraint were removed, the constraint component
$-\psi$
in
$\widetilde P_R$
would cease to exist, while the ‘impressed’ component
$+\psi$
in
$\widetilde P_F$
would remain, creating a fictitious force imbalance. Moreover, the requirement that
$+\psi$
and
$-\psi$
cancel imposes a restriction on
$\widetilde P_F$
that is incompatible with the definition of an impressed force, which must be independent of the constraint. Therefore, the representational freedom has no physical standing.
Once
$P_F$
is fixed by the impressed physics (if any), the reaction pressure
$P_R$
solving (2.7) is uniquely determined up to an additive constant. Once
$P_R$
is solved, the optimal update is
$u_t^\star =-(C+\boldsymbol{\nabla }\!P_R/\rho )$
, and the instantaneous Euler balance
$\rho (\partial _t u+(u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u)=-\boldsymbol{\nabla }(P_F+P_R)$
is recovered.
3.4. Uniqueness of Dirichlet orthogonality
As established above, when
$P_F$
is defined by physically prescribed impressed forces, the decomposition
$p=P_F+P_R$
is already unique. The Dirichlet orthogonality convention discussed below is therefore primarily relevant to the formal decomposition framework of Peters & Ormiston (Reference Peters and Ormiston2025), where additional representational degrees of freedom arise from their harmonic-gauge convention.
Peters & Ormiston (Reference Peters and Ormiston2025) propose the following Dirichlet orthogonality as a convenient gauge-fixing convention for selecting a unique representative of the bookkeeping split
$p=P_F+P_R$
(once a reference/impressed subspace has been chosen)
\begin{equation} \int _\varOmega \boldsymbol{\nabla }\!P_R\boldsymbol{\cdot }\boldsymbol{\nabla }\!P_F\,{\rm d}V = 0. \end{equation}
It is stressed that (3.5) is not a physical restriction on an impressed force defined by external physics; it is a convention that fixes the gauge in the pressure bookkeeping. If
$P_F$
is reserved for physically prescribed impressed potentials, the split is already unique (with
$P_R$
the unique Lagrange multiplier enforcing the constraints) and (3.5) need not hold.
In Appendix A, it is shown that given any provisional solution
$(\widehat P_F,\widehat P_R)$
, the unique orthogonal pair is obtained by a correction
\begin{equation} P_R = \widehat P_R - \sum _{j} \alpha _{\!j} \psi _{\!j},\quad P_F = \widehat P_F + \sum _{j} \alpha _{\!j} \psi _{\!j}. \end{equation}
The coefficients
$\alpha _{\!j}$
are the orthogonal projection of
$\widehat {P}_R$
onto the impressed subspace
$\mathcal{S}_{\textit{imp}} = \text{span}\{\psi _1, \ldots , \psi _m\}$
. We are subtracting this projection from
$P_R$
(making it orthogonal to
$\mathcal{S}_{\textit{imp}}$
) and adding it to
$P_F$
(so the total pressure remains unchanged).
In other words, one can choose:
-
(i) what
$\mathcal{S}_{\textit{imp}} = \text{span}\{\psi _1, \ldots , \psi _m\}$
is. In other words, which pressure modes one may want to impress (e.g. circulation patterns); -
(ii) how much of each mode to impress, typically by specifying boundary conditions or forcing terms.
The Dirichlet orthogonality condition then ensures consistent bookkeeping when decomposing the resulting total pressure
$p$
(which emerges from the full flow physics). Here,
$\mathcal{S}_{\textit{imp}}$
is the ‘signature’ of the forcing mechanism. Dirichlet orthogonality thus labels as forcing (
$P_F$
) everything that has this signature; and labels as constraint (
$P_R$
) everything orthogonal to it (in the Dirichlet sense). We further note that the above is exactly analogous to the Helmholtz–Leray projection operator
$\mathbb P$
operating on any given velocity field
$w$
\begin{align} \mathbb P w \;=\; w-\boldsymbol{\nabla }q,\quad {\nabla} ^2 q = \boldsymbol{\nabla }\!\boldsymbol{\cdot }w \quad \text{in } \quad \varOmega ,\quad \partial _n q = w \boldsymbol{\cdot } n \quad \text{on } \quad \partial \varOmega . \end{align}
Equivalently,
$\mathbb P w$
is the unique vector that satisfies orthogonality
$\int _{\varOmega } \mathbb P w\boldsymbol{\cdot }\boldsymbol{\nabla }\phi \,\mathrm{d}x=0$
. This is a well-known result (See for instance Sohr Reference Sohr2012), but the connection to the impressed/constraint split is new. Appendix B clarifies another point relevant to the projection.
3.5. Physical impressed forces vs. reference (gauge) choices
For the exterior airfoil problem the distinction between physical impressed forces and bookkeeping/reference choices is immediate. The circulation degree of freedom in exterior potential flow resides in the harmonic subspace of the velocity (divergence free, curl free, wall tangent in a multiply connected domain) and cannot be represented as
$\boldsymbol{\nabla }\psi$
of a single-valued scalar pressure potential; hence it cannot be embedded into
$P_F$
as a physical pressure field. By contrast, conservative body-force potentials (e.g. gravity) may legitimately be placed in
$P_F$
as prescribed impressed physics. If one introduces additional reference content in
$P_F$
for diagnostics, then that choice is conventional.
-
(i) Harmonic velocity modes (circulation) are not single-valued pressure gradients. In an exterior domain, circulation enters through the harmonic subspace of the velocity (divergence free, curl free, wall tangent with non-zero loop integral). Such velocity modes are not of the form
$\boldsymbol{\nabla }\psi$
for any single-valued scalar field, so they cannot be represented as a pressure potential in
$P_F$
. In the fixed-time Gauss/Appell step, the circulation parameter
$\varGamma$
appears only through the chosen frozen state
$u(\boldsymbol{\cdot },t)$
.However, the frozen velocity state
$u(\boldsymbol{\cdot };\varGamma )$
determines the convective acceleration
$(u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u$
, which in turn determines the source term in the Poisson–Neumann problem for
$P_R$
. The circulation parameter
$\varGamma$
is therefore reflected in the reaction-pressure field
$P_R$
. For example, a purely circulatory flow
$u=({\varGamma }/{2\pi r})e_\theta$
over a cylinder has convective acceleration
$u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla }u = -\boldsymbol{\nabla }\psi$
with
$\psi = -\varGamma ^2/(8\pi ^2 r^2)$
, yielding
$P_R = \psi$
. Each value of
$\varGamma$
is associated with a specific reaction pressure
$P_R(\boldsymbol{\cdot };\varGamma )$
whose magnitude (the minimised Appellian
$\mathcal{A}^\star$
) varies with
$\varGamma$
. The direct, fixed-time Gauss step does not select
$\varGamma$
(it returns
$u_t^\star = 0$
for every steady state). Gonzalez & Taha (Reference Gonzalez and Taha2022) use the Appellian cost
$\mathcal{A}^\star (\varGamma )$
as a functional for a secondary minimisation over the family of kinematically admissible steady states. -
(ii) Conservative body-force potential. It is legitimate to include a prescribed uniform acceleration (e.g. gravity) through an impressed potential
$P_F(x)=-\rho \,g\boldsymbol{\cdot }x$
, for which
$(1/\rho )\boldsymbol{\nabla }\!P_F=-g$
. The Poisson step then returns
$P_R$
to enforce
$\boldsymbol{\nabla }\boldsymbol{\cdot }u_t=0$
and
$u_t\boldsymbol{\cdot }n=0$
for the frozen
$u$
. -
(iii) Reference pressure for diagnostics (gauge). One may define a reference pressure for interpretation, e.g. subtract the Bernoulli pressure of a chosen reference potential flow in post-processing:
$p\prime := p + \tfrac 12\rho |u_{\textrm {ref}}|^2$
. This is a change of variables (gauge) and should not be interpreted as an impressed force. For a given snapshot
$u$
with no impressed forces, the pressure gradient is entirely the reaction that enforces incompressibility and wall compatibility (the Leray projection of
$-\rho (u\boldsymbol{\cdot }\boldsymbol{\nabla })u$
).
These remarks are particularly relevant to the discussion in Peters & Ormiston (Reference Peters and Ormiston2025) regarding ‘impressed pressure’. In the present framework, the frozen-time Gauss/Appell step returns
\begin{align} u_t^\star = -\left((u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u+\frac 1\rho \boldsymbol{\nabla }\!P_F+\frac 1\rho \boldsymbol{\nabla }\!P_R \right)\!, \end{align}
where
$P_F$
is reserved for prescribed conservative impressed forcing (if any), and
$P_R$
is the reaction (constraint) pressure. Enforcing the acceleration-level constraints
$\boldsymbol{\nabla }\!\boldsymbol{\cdot }u_t^\star =0$
in
$\varOmega$
and
$u_t^\star \!\boldsymbol{\cdot }n=0$
on
$\partial \varOmega$
yields the Poisson–Neumann problem
\begin{align} {\nabla} ^2P_R=-\rho \,\boldsymbol{\nabla }\!\boldsymbol{\cdot } \left((u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u+\frac 1\rho \boldsymbol{\nabla }\!P_F \right)\!, \quad \partial _n P_R=-\rho \left((u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u+\frac 1\rho \boldsymbol{\nabla }\!P_F \right) \boldsymbol{\cdot }n, \end{align}
which determines
$P_R$
uniquely up to an additive constant. A genuinely impressed pressure potential corresponds to an externally prescribed conservative force
$f=-\boldsymbol{\nabla }\varPhi$
, for which one may take
$P_F=\rho \varPhi$
; in the unforced exterior Euler airfoil setting
$P_F\equiv 0$
and the entire pressure is reaction/constraint pressure.
Additionally, since
$\boldsymbol{\nabla }\times \boldsymbol{\nabla }(P_R+P_F)=0$
, the pressure reaction is irrotational and thus does not appear explicitly in the interior vorticity evolution obtained by taking the curl of the momentum equation:
$\omega _t+(u\!\boldsymbol{\cdot }\!\boldsymbol{\nabla })\omega =(\omega \!\boldsymbol{\cdot }\!\boldsymbol{\nabla })u$
(for conservative impressed forcing). We note that this statement concerns the bulk equation; boundary vorticity fluxes can be expressed in terms of tangential pressure gradients (e.g. Lighthill Reference Lighthill1963; Morton Reference Morton1984).
Finally, this viewpoint unifies interpretation and practice. The standard predict–Poisson–correct algorithm is precisely the Gauss–Appell step on a grid, and
$\mathcal{A}^\star$
furnishes a diagnostic that can be monitored in computation: spikes may signal incompatible boundary data, under-resolved features, or an unhelpful choice of impressed/reference content. In reduced-order or data-assisted settings, the choice of
$S_{\textit{imp}}$
allows trusted physics to be injected while the same projection guarantees exact enforcement of incompressibility and wall conditions.
Acknowledgements
Thanks to R. Ormiston, H. Taha, D. Peters and P. Roe for discussions, and for inspiring this work. Special thanks to an anonymous reviewer who helped improve the clarity of sections 3.3 and 3.5.
Declaration of interests
The author reports no conflict of interest.
Appendix A. Uniqueness of Dirichlet orthogonality
Substitute the correction (3.6) into the orthogonality condition (3.5)
\begin{align} 0 = \int _{\varOmega } \boldsymbol{\nabla }\!P_R \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V = \int _{\varOmega } \boldsymbol{\nabla }\left (\widehat {P}_R - \sum _{j=1}^m \alpha _{\!j} \psi _{\!j}\right ) \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V \nonumber \\ = \int _{\varOmega } \boldsymbol{\nabla }\widehat {P}_R \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V - \sum _{j=1}^m \alpha _{\!j} \int _{\varOmega } \boldsymbol{\nabla }\psi _{\!j} \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V. \end{align}
Rearranging gives the
$m \times m$
linear system
\begin{align} \sum _{j=1}^m \alpha _{\!j} \underbrace {\int _{\varOmega } \boldsymbol{\nabla }\psi _{\!j} \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V}_{G_{ij}} = \underbrace {\int _{\varOmega } \boldsymbol{\nabla }\widehat {P}_R \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V}_{b_i}, \quad i = 1, \ldots , m. \end{align}
In matrix form:
$G \boldsymbol{\alpha } = \mathbf{b}$
, where:
-
(i)
$G_{ij} = \int _{\varOmega } \boldsymbol{\nabla }\psi _{\!j} \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V$
is the Gram matrix in the Dirichlet inner product
$\langle p, q \rangle \triangleq \int _{\varOmega } \boldsymbol{\nabla }\!P \boldsymbol{\cdot }\boldsymbol{\nabla }q \, {\rm d}V$
; -
(ii)
$b_i = \int _{\varOmega } \boldsymbol{\nabla }\widehat {P}_R \boldsymbol{\cdot }\boldsymbol{\nabla }\psi _i \, {\rm d}V$
projects the provisional reaction pressure onto each basis function.
Once
$\boldsymbol{\alpha }$
is computed by solving
$G\boldsymbol{\alpha } = \mathbf{b}$
, the unique Dirichlet-orthogonal decomposition (3.6) is achieved.
Appendix B. Rank and projection
Peters & Ormiston (Reference Peters and Ormiston2025) cast multipliers via a finite-dimensional Lagrange-multiplier theorem, tying the number of independent multipliers to the rank of a constraint Jacobian. In a discrete setting this perspective can be helpful. In the PDE setting, however, incompressibility and impermeability impose pointwise constraints. The Leray projector extracts the divergence-free, wall-tangent component, and the reaction pressure is recovered from the gradient component in the decomposition. The velocity space also contains a finite-dimensional harmonic subspace (e.g. circulation modes), which is left unchanged by the projector; see e.g. Sohr (Reference Sohr2012) and the Euler/Appellian equivalence proof in Gonzalez & Taha (Reference Gonzalez and Taha2022).
This viewpoint also makes the analogue of a ‘rank/consistency’ condition explicit. The Neumann Poisson problem for
$P_R$
is solvable if and only if the volume source and boundary flux satisfy the compatibility condition
\begin{align} \int _\varOmega (-\,\boldsymbol{\nabla }\!\boldsymbol{\cdot }C)\,{\rm d}V = \int _{\partial \varOmega } (-\,C\!\boldsymbol{\cdot }n)\,{\rm d}S, \end{align}
which holds identically by the divergence theorem. Uniqueness is up to an arbitrary constant.
Physically, the decomposition expresses the familiar fact that pressure is a reaction field that enforces ideal kinematic constraints without doing work on admissible virtual velocities. Indeed, for any
$v\in V$
\begin{align} \int _\varOmega v\boldsymbol{\cdot }\boldsymbol{\nabla }\!P_R\,{\rm d}V = \int _{\partial \varOmega } P_R\,v\boldsymbol{\cdot }n\,{\rm d}S - \int _\varOmega P_R\,\boldsymbol{\nabla }\!\boldsymbol{\cdot }v\,{\rm d}V = 0. \end{align}
Thus the reaction pressure is
$L^2$
-orthogonal to the space of admissible velocities, exactly as in d’Alembert’s principle, and the ‘multiplier’ is most naturally interpreted as the gradient component required to project
$\rho (u_t+C)$
, rather than as a finite-dimensional rank count.
One additional point is that, unlike in finite-dimensional vector spaces, norms are not equivalent in the function-space setting. Physically, the advective term
$(u \boldsymbol{\cdot }\boldsymbol{\nabla })u$
is unlikely to remain an
$L^2$
function in most Euler solutions. Thus, other weaker norms such as the
$H^1$
-norm are more plausible and have been considered in related works such as Hoffman & Johnson (Reference Hoffman and Johnson2010) and Hoffman et al. (Reference Hoffman, Jansson and Johnson2016) and offer room for additional explorations.
