10. Airflow model#
10.1. What are we calculating?#
Given the incoming wind and a fixed rigid wing, we want the air’s velocity and pressure around its section. The velocity beside the wing is not simply the incoming wind everywhere: the surface redirects the flow.
This chapter develops two-dimensional incompressible viscous Navier–Stokes flow: air keeps approximately constant density, while friction between neighboring layers affects its motion. We then show one numerical way to solve it. The next experiment, A wing in a viscous wind tunnel, uses this grid projection method in a fixed side view, with deliberately increased viscosity and an approximate wing boundary.
Note
Is this different from hydrodynamics? Fluid dynamics covers both gases and liquids. Hydrodynamics often names liquid-flow problems, though it is also used more broadly; aerodynamics names gas-flow applications. These are not competing equations. Navier–Stokes can describe either, with the appropriate material properties and boundaries. Viscous means internal friction is included, not that the fluid must be liquid or thick syrup; see NASA’s Navier–Stokes explanation. Incompressible means we treat density as constant, here uniformly constant. Air is generally compressible, but slow airflows can often use this approximation, as NASA explains for low-speed air.
10.2. How slow is low-speed air?#
Here low speed means slow compared with sound, not slow compared with a person. Let \(V\) be the local flow speed and \(a\) the speed of sound, both in m/s. Their ratio is the Mach number, a number without units:
This is NASA’s Mach-number definition. Near 20°C, sound travels at about 343 m/s in air; see OpenStax’s sound-speed example. Therefore:
15 m/s, or 54 km/h, gives \(M=15/343\approx0.044\).
30 m/s, or 108 km/h, gives \(M=30/343\approx0.087\).
Both are well below the common Mach 0.3 guideline for neglecting compressibility, used for example in NASA’s low-speed wind-tunnel work. At this temperature, Mach 0.3 corresponds to about \(0.3\times343=103\) m/s. It is a guideline, not a sharp cutoff. Constant-density flow is typically appropriate at 15 or 30 m/s when local speeds also stay low: air accelerates around a wing, and temperature changes the sound speed. Those checks concern density changes, not whether the wind feels strong.
Low Mach does not mean laminar flow or absence of separation. Those depend on viscosity, geometry and Reynolds number as well. Nor does it validate this book’s deliberately high-viscosity, Reynolds-100 experiment as a full-scale real-air wing simulation. The applicability of the incompressible equations and the accuracy of our coarse implementation are different questions.
10.3. The physical rules, in order#
Imagine dividing a side view of the air around a rigid wing into small square cells. Each square represents air extending uniformly along the wing span; we calculate no motion in that third direction. We treat air as smoothly distributed material, rather than following each molecule: this is a continuum approximation. Each location has an air velocity and pressure; assigning a quantity throughout space is called a field.
Three rules connect the cells:
Account for mass. Density is mass per volume. With constant density, air cannot accumulate in a fixed cell: total inflow must equal total outflow. Incompressible means this approximation, not that real air can never compress.
Account for acceleration. Acceleration is a change in velocity: speed, direction, or both. Pressure differences, viscous friction, and external forces can cause that change. Newton’s second law says that the net force on a parcel equals its mass times its acceleration.
Respect the wing and the outside wind. Air cannot pass through the solid wing. Viscous air touching its fixed surface has zero velocity. Far away, we supply the incoming wind. These boundary rules are part of the calculation, not decorations added after solving the equations.
Pressure and viscosity have different jobs. Pressure pushes equally in all directions locally; a pressure difference accelerates air toward lower pressure. Viscosity resists neighboring layers sliding at different speeds. Stress is force per area. In our Newtonian approximation, viscous stress is proportional to the rate at which those layers deform. Pressure must adjust throughout the domain so that acceleration does not violate the first rule.
10.4. Unknowns, inputs, and units#
Let \(x,y\) locate a point in metres: rightward and upward, respectively. Let \(t\) be time in seconds. Symbols with an arrow above them are vectors, quantities with size and direction. The unknown \(\vec{u}(x,y,t)=(u_x,u_y)\) is air velocity in m/s, with horizontal and vertical components. The unknown \(p\) is pressure in pascals, Pa = N/m². Their values generally vary across the plane.
Choose constant density \(\rho=1.225\) kg/m³ and dynamic viscosity \(\eta=1.8\times10^{-5}\) Pa s for the examples. These are illustrative air properties, not measurements for every weather condition. The wing experiment uses this density but a much larger viscosity, specified below. Let \(\vec{g}=(0,-9.81)\) m/s² be near-Earth gravitational acceleration. The incoming wind \(\vec{W}\) is a prescribed velocity vector in m/s. The rigid wing’s geometry and angle are prescribed and held fixed during an airflow run.
10.5. Follow one moving parcel#
A small parcel is a portion of air that we follow as it moves, not a fixed grid cell that air passes through. Let its position, velocity, and acceleration be vectors \(\vec{r}_p(t)\), \(\vec{v}_p(t)\), and \(\vec{a}_p(t)\). A derivative measures a rate of change:
Velocity measures how position changes, in m/s. Acceleration measures how velocity changes, in m/s². With fixed parcel mass \(m\) in kg, Newton’s second law is
The pressure-force vector sums the pressure pushes on all sides of the parcel. The viscous-force vector sums friction forces from neighboring air. Each force is measured in newtons. Dividing the net force by mass gives acceleration; acting for a short time \(\Delta t\), it changes velocity by approximately \(\vec{a}_p\Delta t\).
Momentum is \(m\vec{v}_p\); force is its rate of change. For fixed mass, this is the same law as mass times acceleration, not an additional step that converts incoming velocity into force. Air can keep moving at constant velocity with zero net force.
This vector description needs no spatial-derivative shorthand. But the forces depend on the surrounding air, so we cannot solve each parcel independently. Pressure and velocity must also respect mass conservation and the wing boundary.
10.6. From a moving parcel to a fixed grid#
The parcel above explains the physical law; the airflow solver does not store or track individual parcels. It stores samples of a velocity field: a rule giving the air’s velocity at a chosen position and time. Write this field as \(\vec{u}(\vec{r},t)\), where \(\vec{r}\) is the position we ask about and \(t\) is the time. The grid stores horizontal velocity at left and right cell edges, vertical velocity at top and bottom edges, and pressure at cell centers.
In contrast, \(\vec{v}_p(t)\) means the velocity of the particular parcel we are following. Its position is \(\vec{r}_p(t)\). To find that parcel’s velocity, evaluate the field at its current position:
Read this as: the parcel moves with the air velocity at its current position and time. The parentheses on the right mean evaluating a function, not multiplication. The field describes air velocity throughout space; the parcel samples it along one path.
That velocity can change for two reasons. The field can change with time at a fixed location, or the parcel can move into a location with a different velocity. The symbol \(\partial\) means a derivative with the other coordinates held fixed. Thus \(\partial\vec{u}/\partial t\) measures only the first contribution.
For example, suppose steady air speeds up from 5 to 6 m/s over 1 m along its motion. A parcel moving at 5 m/s encounters an acceleration of approximately \(5\times(6-5)/1=5\) m/s² even though velocity at each fixed location is not changing with time.
Write the field’s velocity components as \(u_x,u_y\). Following the parcel gives the vector acceleration
evaluated at the parcel’s position and time. Each spatial derivative is a velocity change per metre; multiplying by the corresponding travel speed gives a velocity change per second. For example, \(u_y\partial\vec{u}/\partial y\) accounts for the velocity change encountered while moving vertically.
The last two terms describe transport, also called advection. They express, at fixed grid locations, the changes already included in the moving parcel’s derivative \(d\vec{v}_p/dt\).
10.7. Writing the two physical balances#
The divergence of velocity measures net outward volume flow per unit volume, in 1/s. Outward contributions are positive, inward contributions negative. Constant-density mass conservation requires
This does not require zero velocity, or equal flow through every edge. It requires their signed total to be zero in every cell.
Pressure differences give a force per volume with two components:
The vector in parentheses is the pressure gradient, measuring pressure increase in each direction, in Pa/m. The minus sign makes the force point toward lower pressure.
For constant viscosity and incompressible flow, let \(\eta\) (Greek eta) be the dynamic viscosity, measuring internal friction, in Pa s. Our illustrative air value is \(\eta=1.8\times10^{-5}\) Pa s. The viscous force per volume is
The sum of second derivatives is called the Laplacian. It compares velocity with neighboring velocities and has units 1/(m s). Both \(\vec{f}_{\mathrm{pressure}}\) and \(\vec{f}_{\mathrm{viscous}}\) have units N/m³: they are forces per volume, unlike the total parcel forces \(\vec{F}\) above. Multiplying acceleration by density gives the same units. Newton’s balance becomes
Read it as density times acceleration = pressure force + viscous force + weight, all per unit volume. These are the incompressible Navier–Stokes equations, not a separate unexplained law. MIT’s Navier–Stokes notes, equations (346)–(347), derive the momentum terms from pressure and Newtonian stress. We write dynamic viscosity explicitly and include gravity.
Still air has a vertical pressure variation that supports its weight, called hydrostatic pressure. Subtracting that known background from pressure removes the matching fluid gravity term consistently. For the algorithm below, \(p\) denotes the remaining pressure and that background has been subtracted; only the remaining pressure drives the flow update.
10.8. What the boundaries supply#
A boundary condition specifies behavior at the edge or a surface of the calculation. Let \(\vec{r}\) be a point on the rigid wing’s surface in metres, and \(\vec{v}\) its velocity in m/s. The derivative \(d/dt\) follows that same material point, unlike a derivative at a fixed location. At the wing surface, viscous no-slip flow requires
For this fixed wing, \(\vec{v}\) is zero: air directly touching its surface must be stationary too. The normal component, perpendicular to the surface, prevents penetration. The tangential component, along it, prevents slipping. Air moving tangentially at 0.10 m/s directly at a stationary no-slip wall would violate that condition. MIT’s notes state the no-slip condition in Section 13.2.
Far away, specify incoming \(\vec{W}\); allow disturbed air to leave through an appropriate downstream boundary. Also specify initial velocities and hold the wing geometry fixed. A closed domain needs a pressure reference; an open domain may prescribe pressure at an outlet. Boundary choices affect the result. Tunnel walls need their own boundary rules; the experiment’s slip outer walls differ from its approximately no-slip wing.
10.9. How one numerical solution works#
Spatial discretization replaces continuous fields with finitely many stored samples. Here is the fluid-specific pressure-projection method, following the approach developed by Chorin and the grid construction in Bridson and Müller-Fischer’s fluid notes, Chapters 2–4. Projection means correcting a provisional velocity so it satisfies the volume balance. This is the algorithm family used in the next wing experiment. The experiment’s specific two-dimensional grid, boundaries, viscosity and stopping tolerance are stated on its page.
Use a regular grid of square cells with spacing \(h\) in metres. Store pressure at their centers and the perpendicular velocity component at each edge. This arrangement is called a staggered grid: pressure and velocity are stored at different locations. The edge velocity directly measures flow between neighboring cells.
Let \(\Delta t\) be a time step in seconds; \(\Delta\) means a change. Let \(\vec{u}^n\) be the stored velocity before a step; \(n\) counts time steps. A star labels a provisional value, not a multiplication. Let \(\nu=\eta/\rho\) be kinematic viscosity in m²/s; our choices give approximately \(1.47\times10^{-5}\) m²/s.
First predict motion without the unknown pressure. For a simple first-order time update, transport momentum and apply viscosity:
The minus transport term moves the stored velocity pattern with the air; viscosity smooths differences between neighboring velocities. For a concrete spatial approximation, estimate transport derivatives using values on the upstream side of the flow, called upwind differences, with interpolation to the needed edges. For each velocity component \(f\) in m/s, let \(f_L,f_0,f_R\) be its values at three equally spaced locations along one coordinate. Let \(w\) be the transporting velocity in that coordinate, also in m/s. Its upwind transport term is
Use the analogous difference along each active coordinate and add the terms. In the wing experiment, transporting horizontal velocity by vertical velocity requires the latter at the edge storing horizontal velocity; it is obtained by averaging the four surrounding vertical-velocity samples, and conversely for vertical momentum. First-order upwinding introduces numerical diffusion in addition to the selected physical viscosity. The second derivative in that direction is approximated by
Add the corresponding contribution from the other coordinate for the Laplacian. Explicit transport and viscosity need sufficiently small time steps; this deliberately simple scheme is not a claim of accuracy at any chosen grid size. More accurate transport and implicit viscosity are possible.
Next find the pressure correction. We want a new velocity \(\vec{u}^{n+1}\) that has zero divergence. Pressure changes the provisional velocity by
Take the divergence of this equation, require the result to vanish, and rearrange. The unknown pressure must satisfy
This is a Poisson equation, which links each pressure to neighboring pressures and the known local volume imbalance. It is how the computer finds pressure, rather than choosing it independently at each location. Use edge-flow differences for divergence and center-pressure differences for the gradient. These must be consistent: applying divergence to the discrete pressure gradient produces the discrete pressure equation.
There is one equation per fluid cell. They form a linear system: many simultaneous equations whose unknowns are the cell pressures. Solve them together, then update the edge velocities using their pressure differences. Solid-wall normal velocities enter both the imbalance and the pressure boundary equations; they must not be changed incompatibly afterward. Tangential no-slip belongs to the viscous boundary treatment. A curved wing needs additional boundary geometry handling, not just this regular-grid recipe; the experiment approximates that surface with blocked cells.
10.10. A small calculation from imbalance to result#
To isolate the pressure step, take two neighboring square cells, each \(h=0.10\) m wide, inside stationary closed walls. Their only shared edge has provisional rightward velocity \(U^*=0.10\) m/s. All exterior edge velocities are zero. This is an illustrative predictor output, not a complete wing-flow simulation or a steady incompressible starting state. Use \(\rho=1.225\) kg/m³ and \(\Delta t=0.01\) s.
Each cell has area \(h^2=0.01\) m², and the shared edge has length \(h=0.10\) m. Flow across that edge per unit wing span is \(U^*h=0.01\) m²/s from the left cell to the right. Equivalently, a 1 m span would have 0.01 m³/s passing between cells of volume 0.01 m³. The left cell loses air; the right gains it. Dividing signed outward flow per unit span by cell area gives divergences
Here \(d_L,d_R\) label the two known divergences, and \(p_L,p_R\) the two unknown pressures in Pa. The walls keep their normal velocity zero, so their pressure-gradient contributions are zero in this example. Each cell’s pressure equation has just one neighbor:
Both equations give \(p_R-p_L=1.225\) Pa. Only the difference is determined: adding the same constant to both pressures changes no force. Choose zero mean as the pressure reference. Then \(p_L=-0.6125\) Pa and \(p_R=0.6125\) Pa, measured relative to that reference, not negative absolute atmospheric pressure.
Pressure rises to the right by \(1.225/0.10=12.25\) Pa/m. It causes leftward acceleration \(-12.25/1.225=-10\) m/s². Over the chosen time step, the shared edge becomes
The wall edges remain zero. Both cell divergences are now zero too. We supplied an imbalanced provisional velocity, solved two coupled pressure equations, and obtained a compatible velocity. In flow around a wing, projection removes the imbalance but need not remove all motion: air can continue circulating through many cell edges.
10.11. Seeing flow with passive tracers#
Once the velocity field is calculated, we can follow marked parcels without letting those markers change the flow. These are passive tracers. They have no mass or force in our model and no independent random drift. Their color is a label, not a measured pressure or speed.
Let \(\vec{X}(t)\) be one tracer’s position in metres. The already calculated velocity \(\vec{u}\) is evaluated at that position. The tracer follows
The \(d/dt\) here follows the marker, rather than holding its location fixed. If the local velocity stays \((2,-0.5)\) m/s for 0.20 s, a marker moves \((0.40,-0.10)\) m. The plotted history of one moving marker is a pathline. A streamline instead follows the directions of the velocity field at one frozen instant; the two coincide only when the flow is steady. The experiment draws pathline histories.
We need velocities between grid edges. Bilinear interpolation weights the four closest stored values according to distance, once for each velocity component. For example, halfway between four samples, their weights are all one quarter. This interpolation does not make the velocity exactly divergence-free between samples or guarantee impermeability at the mask.
For a numerical step, let \(\vec{X}^n\) be the marker’s current position and \(\vec{X}_m\) a provisional position halfway through the same flow time step \(\Delta t\). First estimate
Use the average of old and new velocities at this middle position for the full displacement:
This is an explicit midpoint-style tracer update, using linear time interpolation of the two calculated fields. It is second order for a smooth, accurately supplied velocity, but the experiment’s first-order flow solver and coarse boundaries still limit the overall accuracy. Its code retires a tracer that enters a blocked grid cell rather than drawing it through the wing or bending its path by hand. New markers enter at the inlet; finite trails remember recent positions, not a permanent steady streamline.
Real visible smoke also spreads, can settle, and can rise when heated. For a dilute passive concentration, let \(c\) be smoke mass per volume in kg/m³, \(\kappa\) a diffusivity in m²/s, and \(S\) a source in kg/(m³ s). A possible additional model is
The time derivative and velocity-weighted spatial derivatives describe transport of concentration; diffusion spreads it and the source adds it. Neither this concentration equation nor heat, buoyancy, settling or scattering is implemented by the wing experiment. Calling its markers “smoke trails” describes a visual resemblance, not a smoke model. The fluid/tracer distinction follows Bridson and Müller-Fischer’s fluid notes, which treat passive marker advection separately from buoyancy.
10.12. Which version the wing experiment implements#
The next experiment implements the two-dimensional equations above. Its code names the horizontal and vertical velocity components \(u,v\), corresponding to \(u_x,u_y\) here, so \(\vec{u}=(u,v)\). It calculates no span flow or tip vortices.
In three dimensions, the same balances gain a third position coordinate, a third velocity component, and derivatives in that direction. The two-dimensional model assumes those span variations and span motion are absent.
For feasibility on a coarse browser grid, the wing experiment uses \(\nu=0.015\) m²/s, about a thousand times the illustrative air value, with speed 1 m/s and chord 1.5 m. This is a deliberately viscous model fluid, not realistic airplane air. Its pressure is relative to the outlet reference, and hydrostatic pressure has been removed. No gravity acts on markers.
The wing is a solid-cell mask: grid cells with centers inside its rotated section are blocked. The drawing uses a smooth rendering of that underlying section; the fluid boundary is still a stair-step approximation on the grid. Every velocity edge between fluid and solid has normal speed zero. For viscosity near a wing wall, reflect the tangential velocity into a ghost sample, a numerical value beyond the wall, rather than an extra fluid parcel. If \(f_0\) is the neighboring tangential fluid speed, choose
The zero midpoint approximates no-slip at the mask wall. This reflected sample goes into the centered viscous difference, while transport uses the blocked edge’s zero value. Normal-component wall samples are zero. This is an approximate stair-step boundary treatment, not a resolved smooth-wing boundary layer. Bilinear tracer sampling can still leak into a blocked cell; retiring such a marker is a drawing safeguard, not a change to this flow law.
At the top and bottom of the tunnel, normal speed is zero, but tangential ghost samples equal the adjacent fluid value. This gives zero tangential derivative: slip walls, a different condition from the no-slip wing. The inlet supplies horizontal speed 1 m/s and vertical speed zero. At the outlet, the predictor copies the adjacent velocity (zero normal derivative); pressure is set to zero at the outlet edge. Projection adjusts outgoing normal speed. The condition is too simple for substantial flow entering backward from outside the domain.
Here is the actual discrete pressure system behind the experiment. Let \(i,j\) count column and row; \(p_{i,j}\) is a cell-center pressure and \(d_{i,j}\) its provisional outward volume imbalance per cell volume, in 1/s. Face differences give
Let \(\mathcal N\) be the set of neighboring fluid cells, and \(q\) label one member’s pressure \(p_q\). Multiplying the discrete Poisson equation by \(-h^2\) gives the interior row
Fixed-normal-speed edges add no unknown pressure difference. A zero-pressure outlet is half a cell from the last center, so its contribution on the left is \(2p_{i,j}\). The matching outlet velocity correction uses \((0-p_{i,j})/(h/2)\) as its pressure gradient. Using the same half-cell distance in both operations preserves the discrete volume balance.
These rows form a sparse symmetric positive linear system: diagonal entries count fluid neighbors, with two added at the outlet, and neighboring off-diagonal entries are -1. The experiment uses conjugate gradients, an iterative method for such systems, with the diagonal as a preconditioner (a scaling that helps the iterations). The pressure residual is the right-hand side minus the matrix applied to the current pressure estimate. Multiplying its magnitude by \(\Delta t/(\rho h^2)\) gives the remaining divergence magnitude. This is what the stopping test measures. The structure follows Bridson and Müller-Fischer’s fluid notes, Chapters 4 and 5; it does not justify an arbitrary chosen pressure field.
The code targets maximum residual-equivalent divergence of \(10^{-5}\) 1/s and permits at most 800 iterations. It checks the actual corrected maximum cell divergence against \(2\times10^{-5}\) 1/s and stops with an error if the bound is missed. Prescribed/blocked normal speeds are never overwritten after projection. The interface displays this actual residual.
Let \(M_u,M_v\) be the largest absolute component speeds in m/s. The explicit time step is at most 0.012 s and is shortened to obey
The expression combines transport and viscous rates; the conservative factor helps accommodate the reflected boundary stencils. It is a stability guard, not an accuracy certificate. The display’s 96 by 48 cells are too coarse for real-air boundary layers. Three-grid comparisons expose sensitivity rather than establish convergence. Changing the wing angle resets the mask and projected initial flow, not a moving-boundary wing calculation. See A wing in a viscous wind tunnel for the controls, geometry, numerical checks and precisely what not to infer.
10.13. Repeat the airflow step#
Repeat prediction and projection as time advances, keeping the rigid wing and tunnel boundaries fixed. Changing the angle starts a new run with a new fixed geometry; it is not a moving-wing calculation.
Near a viscous surface, velocity changes through a thin boundary layer. The disturbed air downstream is the wake. Separation means flow peels away from the surface. These are consequences a sufficiently resolved viscous calculation may represent, not extra forces specified by naming them. A coarse calculation is not automatically accurate; high-speed or high-Reynolds-number flow may need finer resolution and additional modeling.
10.14. Separation and turbulence: law versus resolution#
Does this model include flow detaching from the back of a wing? The viscous Navier–Stokes equations permit separation without a separate detachment switch. At a stationary wall, no-slip and viscosity create a low-speed boundary layer. If pressure rises along the flow direction, called an adverse pressure gradient, the pressure force opposes motion. Near-wall flow can slow, reverse and leave the surface, forming a separated shear layer and sometimes recirculation. NASA’s boundary-layer explanation connects the slow near-wall layer, pressure changes and separation.
The full three-dimensional unsteady equations can also describe turbulence, irregular interacting flow fluctuations across many sizes. Calculating it directly requires resolving those sizes and sufficiently short time steps; otherwise an additional turbulence model is needed. Merely writing Navier–Stokes, drawing curling traces or obtaining a small pressure residual does not establish that turbulent behavior is resolved.
The wing experiment has no turbulence model, no span motion and no resolved real-air boundary layer. Its high model viscosity, first-order numerical diffusion and coarse stair-step wall strongly alter the result. Local reverse flow, if measured, would demonstrate recirculation in that numerical model, not a validated real-wing separation point or stall angle. Its experiment page distinguishes tested behavior from effects the continuum law permits.
10.15. From pressure to one force, then lift and drag#
Is air pushing part of this calculation? Yes: the pressure gradient already accelerates air in the momentum update. The computed pressure also lets us estimate the force air applies to the fixed wing. The wing does not move under that load in this experiment.
Pressure acts perpendicular to each surface piece. In this two-dimensional model, report force per unit span, in N/m, not a force on a whole finite wing. Let \(p\) be relative pressure in Pa, \(\Delta s\) a small surface length in metres, and \(\vec{n}\) its outward unit normal, pointing from the solid into the fluid. The normal has no units. Let \(\Delta\vec{f}_p\) be that piece’s pressure-force contribution per unit span. Pressure pushes opposite the outward normal:
The sum adds vectors around the entire closed section, including leading and trailing edges. Its units are Pa times m = N/m. This one resultant is not two independent forces. Let \(\vec{e}_D\) be the unit direction of incoming flow, and \(\vec{e}_L\) the perpendicular unit direction chosen upward. The dot product takes a vector’s component along a direction. Define signed pressure drag \(d_p\) and pressure lift \(l_p\), both in N/m, by
Here incoming air moves rightward, so these are simply the rightward and upward components. A resultant (0.20, 0.60) N/m has pressure drag 0.20 N/m, pressure lift 0.60 N/m and magnitude about 0.632 N/m. On a horizontal 0.10 m piece with upward normal, relative pressure -3 Pa contributes 0.30 N/m upward. These are illustrative calculations, not measured wing performance. NASA’s pressure-to-force explanation describes summing surface contributions and resolving the single force along and across flow.
Negative relative pressure means below the reference, not negative absolute pressure. Its outward arrow is a signed contribution relative to that reference, not an absolute pulling force. A constant pressure shift changes local arrows and colors but cancels from the resultant on a closed surface: the sum of outward normals weighted by surface length is zero. The experiment uses pressure relative to its zero-pressure outlet, with the hydrostatic background removed, not a buoyancy calculation.
These arrows are pressure-only. Full viscous loading also includes surface viscous stress, including shear, which these readouts omit even though viscosity affects the calculated flow. The experiment sums pressure on its stair-step numerical boundary using adjacent fluid-cell values; its smooth drawing is not the integration surface. Arrows are scaled display guides, not wing displacement. Neither this estimate nor a small pressure residual certifies real-wing lift, drag or stall accuracy.