Which came first: the chicken or the egg? velocity or pressure?

Pressure field seems to be the quick answer to almost any question in aerodynamics (and fluid dynamics), from the cause of the lift "force" on an airfoil and wing tip vortices, to more specific aspects such as fluid separation or the generation of vorticity on a surface. For example, the term "adverse pressure gradient" (it sounds so smart) is often used by those trying to justify the cause of fluid separation behind an aerodynamic body. According to a recent survey conducted in a Linkedin CFD group (see Fig. 5), it is clear that two-thirds (36/53) of the participants believe that there is no direct cause-effect between velocity and pressure fields, since both are coupled, while the remaining one-third (17/53, including me) mark a causal in one or the other direction (velocity causes pressure or pressure causes velocity). With such disagreement, this uncertainty must be considered an open question. Therefore, in this short blog post, I will try to justify, from both a computational and a physical point of view, why the pressure field must be considered only as a consequence, and not as a cause of these phenomena.

Animation 1: Full-scale light sport aircraft (LSA) Halcón II near stall condition by Lattice-Boltzmann method simulation (Horizontec-YUMA Engineering; C. Pimentel, 2017). More details: YUMA Engineering - Facebook - Halcón II

Computational perspective

Nowadays, mesh-based approximations of the Navier-Stokes equations (NSE) are the most popular way to solve fluid flow transport phenomena, even for compressible cases where relevant variations of flow density and temperatures may be present. However, most of such Eulerian methods are based on several analytical and semi-empirical models and numerical techniques, riddled with assumptions and simplifications (not always well justified) that try to close the system of equations in more than a controversial way. Even the purest method (direct numerical simulation; DNS) suffers from this lack of consistency, since it is not capable of approximating simulations of low fluid viscosity (or Reynolds numbers tending to infinity), which seems to be more than a computational resource problem (i.e., it does not solve under an inviscid generation of vorticity theory and Euler eqs.). For the incompressible case, the momentum eq. leads to the velocity field, but since the continuity eq. does not include the pressure term, both mass and momentum conservation eqs. must be combined into one (Poisson eq.) to approximate the pressure term. Note that at this point the pressure solution depends on a velocity field obtained by guessing another pressure! This mathematical intrinsicity provokes a coupled velocity-pressure system of equations, which must be approximated by a pressure-correction method (SIMPLE, PISO, etc.). At this point, it is generally assumed that both velocity and pressure fields are causes and at the same time consequences of each other, since both are numerically coupled.

Fig. 1 Complexity of the Poisson equation, derived from momentum and mass conservation equations. Source: MIT's presentation

In contrast to NS-based solutions, the popular Lattice-Boltzmann Method (LBM) solves fluid dynamics from a mesoscale perspective (an intermediate level between molecular and continuum media) based on the Boltzmann transport equation. Some of its main advantages are described next: it solves a first-order partial differential equation (instead of a second-order in NS), convection becomes simple advection, and, most important for the present discussion, pressure is obtained directly from the state equation (which involves velocity distribution functions (v.d.f.): f_i(x,t)). Another advantage in terms of preprocessing complexity is that it does not implement meshes in the strict sense, since it uses Cartesian lattices (or regular grids) for the domain discretization, where streaming and collisions between the pseudo-particles (a set of atoms or molecules) are performed, exactly conserving mass and linear momentum at each time step. To obtain the density (ρ) at each node, the streaming step must be performed*, which inherently involves velocity calculations for the pseudo-particles in each direction. Furthermore, since in LBM the pressure field (p) is directly proportional to the density field (ρ) times the square of the speed of sound (c_s^2), or mathematically: p=ρ*c_s^2 (based on the ideal gas law), where the density depends on the v.d.f. for each velocity direction within each individual piece of the lattice. At this point, it seems clear that, at least on the mesoscale level, the propagation velocity, calculated after collisions of the pseudo-particles causes a variation in density and then in pressure, so there is no direct velocity-pressure coupling.

Fig. 2 Some schemes for 2D and 3D in the Lattice-Boltzmann Method (LBM). Source: LBM chapter

In the velocity-vorticity formulation (i.e., vortex methods) of the incompressible NSE (i-NSE), the pressure (p) field drops out when the rotational operator is applied to both sides of the velocity-pressure momentum equation, so the flow solution depends only on both the flow velocity (u) and vorticity (ω) fields (see Fig. 3). However, the pressure field can be reconstructed a posteriori via the stream function (ψ). It should be remembered that the main advantage of the Boundary Element Method (BEM; linked naturally to vortex methods) is that it allows to solve a flow problem only by obtaining its variables on the discretized surface, avoiding the use of the typical three-dimensional domain discretization (as in the Eulerian description), making use of one of the most powerful mathematical tools: the Green's theorem; in the end, to obtain the resultant force, moments and pressure distribution acting on the object, it is not necessary to perform additional calculations. Furthermore, as in any dynamical system, fluid dynamics must be conditioned to obey the same principles: by definition, a decoupled variable does not affect either the solution or any other variable, so the pressure field does not affect either the velocity (or vorticity) field or the flow solution. From this particular Lagrangian approach it can be concluded that the pressure field does not cause anything (velocity or vorticity, forces and moments, flow separation or any other physical phenomenon), but is only a consequence of the numerical solution of the equations of motion. Velocity and vorticity cause everything! [1].

Fig. 3 From velocity-pressure (blue) to velocity-vorticity (red) formulations of the momentum equation(s).

Physical perspective

Typically, the Bernoulli equation, which is ultimately a simplified version of the i-NSE, states that the kinetic, potential, and internal energies remain constant along a streamline and it can be applied to a set of them within a tube. The first of these energies, transformed into its dynamic pressure version, is uniquely defined by only two parameters: the flow density (ρ) and its velocity (v): q=0.5*ρ*v^2. Since it comes from the definition of kinetic energy (E_k), such an equation is quite similar to E_k=0.5*m*v^2, where m is the mass of an object (e.g. a molecule or a set of them). In the last equation it is clear that, for a constant mass, the kinetic energy depends only on the square of the velocity of the object; by analogy, the dynamic pressure (for a constant flow density) depends only on the same variable (velocity). From this simple analysis, we can conclude that flow velocity causes dynamic pressure (just as velocity causes kinetic energy), but not the other way around. But what about static pressure? or, in other words, where does such pressure come from if the flow velocity (at the macroscale) is zero? The answer comes from the kinetic theory of gases, where fluid particles (atoms or molecules) are constantly in motion, colliding with each other and with surfaces, causing static pressure. At the microscale level, individual particle velocities do not depend on a constant, but on different velocities determined by the intensity of the collisions with other particles before hitting a surface (e.g. water in a manometer). Thus, the measured static pressure is a direct consequence of the number of surrounding particles (density) and the impact intensity (due to particle velocities). Once again, (static) pressure seems to be a consequence, but not the cause of velocity, not only at the macroscopic, but even at the microscopic level.  

Animation 2: Monoatomic particle collision simulation (neither mass nor energy is transmitted through the walls). Source: Elastic collision

Based on the previous animation, I will now describe why an instantaneous increase in volume (and thus lower static pressure) cannot physically justify any particle acceleration (or change in velocity). This process is analogous to what happens in a clogged syringe.

Fig. 4 A change in the (static) pressure cannot justify any change in instantaneous particle velocities. Source: own.

a) At the initial time (t_0), a set of 4 particles is contained in a volume (V_0); some of them (3; yellow) hit the walls, where a static pressure (p_0) is measured.

b) An instantaneous increase in volume is applied just before the marked (red) particle hits the right wall. At this point, the measured static pressure (p_0*) is exactly the same as in the previous case. Since the total mass is conserved, the density (ρ_0*) decreases. Note that the magnitude of the velocity (v) of such a particle (and the others) remains the same (typically hundreds of m/s!); a physical increase in velocity must imply a decrease in the mass (m) of the particle in order to preserve its linear momentum!, hence no "micro-suction" effect can be physically justified.

c) All particles have naturally evolved (hypothetically without collisions between them) in time and populate the generated void, where (total) density (ρ_0+Δt) and volume (V_0+Δt) remain. Since the marked (red) particle has not collided from the previous case (at t_0*), so it hits the right wall with the same velocity (v). Nevertheless, the measured static pressure (p_0+Δt) is lower than in the previous instant (at t_0), because the density has also decreased, so there is less number of collisions with walls at a given moment. 

Fig. 5 Results of the survey on the relationship between velocity and pressure fields.

It should be emphasized that the previous simplified example can be applied in the same way to a more populated volume, where a new collision scheme (hence a different instantaneous velocity field) will preserve the total linear momentum, so that the impact intensity on the walls (hence reaction forces, hence pressure) will be proportional to the previous instant (at t_0), especially from an averaging perspective, where a huge number of particles are moving and colliding around and above the speed of sound.

From the previous explanation, it is clear that no change in pressure justifies a change in the instantaneous velocity field, or in other words, the pressure field does not cause the velocity field at the microscopic scale. Since such a phenomenon does not occur at the microscale level, it is not expected to occur at a larger scale.

* some LBM algorithms change the order of such steps (from streaming-collision to collision-streaming).

[1] PDF presentation: VortoNeX project (meshless CFD patent) (youtube.com)