7. Pressure–Velocity Coupling and Boundary Conditions

Introduction and Context

The numerical solution of the incompressible Navier–Stokes equations is governed by a difficulty that does not arise in the same form for compressible flow: pressure has no separate evolution equation. In incompressible motion, pressure acts primarily as a Lagrange multiplier that adjusts the velocity field so that continuity is satisfied. Consequently, pressure and velocity must be solved as a coupled system.

For a Newtonian incompressible fluid of constant density, the governing equations are

\[

\nabla\cdot \mathbf{u}=0

\]

and

\[

\frac{\partial \mathbf{u}}{\partial t}+\nabla\cdot(\mathbf{u}\mathbf{u})=-\frac{1}{\rho}\nabla p+\nabla\cdot(\nu\nabla\mathbf{u})+\mathbf{f}.

\]

The pressure gradient appears in each momentum equation, whereas pressure itself is determined indirectly through continuity and the boundary conditions. A discretization may therefore produce apparently reasonable momentum equations while still generating an unstable, non-conservative, or physically inconsistent flow field if pressure–velocity coupling is treated incorrectly.

This chapter develops the principal arrangements and algorithms used to address this issue. It also explains how initial and boundary conditions must be formulated, how staggered and collocated grids influence pressure–velocity interpolation, and how SIMPLE-family, PISO, projection, and fully coupled methods differ in practice.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain why pressure and velocity are intrinsically coupled in incompressible flow.

2. Distinguish Dirichlet, Neumann, Robin, symmetry, periodic, wall, inlet, outlet, and pressure-reference conditions.

3. Formulate physically consistent initial and boundary conditions for common flow problems.

4. Compare staggered and collocated arrangements, including the origin and prevention of pressure–velocity decoupling.

5. Derive the logic of pressure-correction methods and explain the SIMPLE, SIMPLEC, PISO, projection, and coupled algorithms.

6. Select an appropriate coupling strategy for steady, transient, laminar, turbulent, and industrial simulations.

7. Diagnose common convergence and boundary-condition failures.

8. Assess the role of modern matrix solvers, adaptive time stepping, and automated CFD workflows.

7.1 Why incompressible pressure is a constraint variable

In a compressible formulation, pressure can often be linked to density and energy through an equation of state, and pressure disturbances propagate through the fluid at a finite acoustic speed. In the incompressible limit, density is fixed and the continuity equation becomes a kinematic constraint. Pressure is whatever field is required to enforce that constraint.

Consider a discretized momentum equation for a velocity component:

\[

a_P u_P=\sum_N a_Nu_N+b_u-(p_E-p_W)A_P.

\]

The velocity depends on pressure differences. At the same time, the face mass fluxes used in the discretized continuity equation depend on velocity. The pressure field must therefore be adjusted until the fluxes satisfy

\[

\sum_f \dot m_f=0

\]

for every control volume. Solving momentum alone does not guarantee mass conservation; solving a pressure equation alone does not determine momentum. The two equations form a coupled algebraic system.

A pressure field is also defined only up to an additive constant in a closed incompressible domain. If p is a solution, p+C produces the same pressure gradient and the same velocity. One pressure value, or the volume-average pressure, must therefore be prescribed as a reference. This is a mathematical gauge condition rather than an additional physical force.

7.2 Initial and boundary conditions: physical formulation before numerical implementation

Initial conditions

A transient problem requires the initial velocity and, where relevant, temperature, turbulence variables, species concentrations, and other transported quantities. The initial field need not satisfy the final solution, but it should be physically admissible and should not create avoidable numerical transients.

A uniform quiescent field is appropriate for a fluid initially at rest. For a suddenly started lid-driven cavity, however, imposing the lid velocity abruptly creates an impulsive start and a sharp initial shear layer. This may be useful for a benchmark but is not equivalent to gradual acceleration. In engineering calculations, ramping the inlet or wall velocity over several time steps can reduce nonphysical startup oscillations.

For incompressible flow, the initial velocity should preferably be projected onto a divergence-free space, or corrected during the first pressure solve. If the initial field has large continuity errors, the first pressure correction may be excessively large and can destabilize the calculation.

Inlet conditions

At a solid inlet, velocity is usually prescribed, making it a Dirichlet condition. A fully developed profile, a uniform profile, or a measured profile may be used depending on the physical situation. For internal turbulent flow, specifying only the mean velocity is insufficient when a turbulence model requires turbulence kinetic energy and dissipation or specific dissipation rate. These quantities should be consistent with turbulence intensity and length-scale estimates.

For an incompressible flow with prescribed velocity, the total mass flow rate must be compatible with the outlet condition. If the inlet velocity is fixed over an area A, the imposed rate is

\[

\dot m=\rho\int_A \mathbf{u}\cdot\mathbf{n}\,dA.

\]

Over-specifying both velocity and pressure at the same boundary generally makes the problem mathematically inconsistent unless one condition is used only as a reference or applied to a different characteristic variable.

Outlet conditions

Outlet boundaries require particular care because information should generally leave the domain with minimal artificial reflection. Common choices include prescribed pressure, zero normal gradient, convective or advective outflow, and fully developed conditions.

A pressure outlet often specifies static pressure while allowing the velocity to be determined from the interior solution. A zero-gradient condition for velocity is convenient, but it is not universally physical. It becomes questionable when strong recirculation, flow reversal, separation, or vortices reach the outlet. The computational domain should normally be extended so that the outlet is located in a region where the flow is sufficiently developed.

A convective outlet condition may be written schematically as

\[

\frac{\partial \phi}{\partial t}+U_c\frac{\partial \phi}{\partial n}=0,

\]

where Uc is an estimated convection speed. Such a condition is useful for predominantly downstream transport but can be inaccurate when disturbances travel upstream or when the exit flow is strongly three-dimensional.

Solid walls

For a stationary impermeable no-slip wall,

\[

\mathbf{u}=0.

\]

For a moving wall, the tangential velocity equals the wall velocity and the normal velocity satisfies impermeability. At high Reynolds number, wall functions may replace direct resolution of the viscous sublayer; their use must be consistent with the near-wall mesh and turbulence model. Applying a wall-function treatment to a mesh designed for low-Reynolds-number resolution can produce incorrect wall shear and heat transfer.

The pressure normal derivative at a no-penetration wall follows from the normal momentum equation. It is not generally correct to impose an arbitrary fixed pressure at every wall. In a projection method, a homogeneous Neumann condition for the pressure correction commonly arises when the normal velocity is prescribed, but the precise discrete condition depends on the momentum equation and time integration.

Symmetry and periodic boundaries

At a symmetry plane, normal velocity is zero and normal gradients of tangential velocity and scalar variables vanish. In vector form, the normal component of velocity is zero and the tangential derivative of the tangential velocity is zero. A symmetry condition should not be used where the geometry or flow contains asymmetric separation, swirl, or instability.

Periodic boundaries identify corresponding faces and require equality of appropriate variables or fluxes, possibly with a prescribed pressure drop. They are particularly useful for fully developed channels, repeating porous structures, and homogeneous turbulence computations.

Pressure and velocity boundary compatibility

A robust rule is to specify enough independent information to make the differential problem well posed, but not more. For a pressure-driven channel, a pressure difference or pressure gradient and no-slip walls are appropriate; the flow rate is then an outcome. For a flow-rate-driven channel, the flow rate or inlet velocity is prescribed and an outlet pressure reference is used. Imposing both independently may over-constrain the system.

The global compatibility condition is also important. In a closed incompressible domain, the net prescribed mass flux must be zero. In an open domain, total inflow must equal total outflow at each converged steady state. A pressure correction equation with incompatible source terms has no physically meaningful solution.

7.3 Grid arrangements and the prevention of decoupling

Staggered arrangement

In the classical staggered arrangement, pressure is stored at cell centres, while the normal velocity component is stored at the corresponding cell faces. In a two-dimensional Cartesian grid, u is placed on vertical faces and v on horizontal faces. The pressure difference driving a face velocity is therefore directly available between adjacent pressure cells.

This arrangement has an important advantage: the pressure gradient and face mass flux are naturally connected. It strongly suppresses checkerboard pressure fields, in which neighbouring pressure values alternate while producing a misleadingly small discrete pressure gradient at velocity locations.

Its disadvantages are increased bookkeeping, more complicated data structures on unstructured meshes, and greater difficulty when implementing general boundary conditions or multiphysics coupling.

Collocated arrangement

In a collocated arrangement, pressure and all velocity components are stored at cell centres. This is convenient for arbitrary polyhedral meshes and software development, but face velocities and face mass fluxes must be reconstructed. Simple linear interpolation can permit pressure checkerboarding because the pressure difference may disappear from the interpolated face flux in an alternating pressure pattern.

The widely used remedy is Rhie–Chow-type momentum interpolation. The face velocity is constructed from interpolated cell velocities together with a pressure-difference term based on neighbouring pressure values and momentum coefficients. The correction is designed to affect the mass flux without introducing an inconsistent artificial pressure diffusion. On unstructured meshes, the same principle is implemented through a pressure-weighted or momentum-consistent face-flux interpolation.

Collocated schemes also require careful treatment of non-orthogonal meshes. The pressure gradient is decomposed into an orthogonal contribution, treated implicitly when possible, and a non-orthogonal correction, often treated explicitly or through deferred correction. Excessive non-orthogonality can reduce accuracy and slow convergence.

7.4 Pressure-correction formulation

Let the discretized momentum equation be written symbolically as

\[

A_u\mathbf{u}=\mathbf{H}_u-G_up,

\]

where Au contains the diagonal and neighbour coefficients, Hu contains convection, diffusion, source, and boundary contributions, and Gu represents the discrete pressure gradient. Formally,

\[

\mathbf{u}=A_u^{-1}\mathbf{H}_u-A_u^{-1}G_up.

\]

Substitution into the discrete continuity equation gives a pressure equation. Direct formation of the exact inverse is impractical, so pressure-correction algorithms approximate the inverse using the diagonal momentum coefficients or related approximations.

Suppose p and u are the current pressure and velocity estimates. Define corrections by

\[

p=p^+p', \qquad \mathbf{u}=\mathbf{u}^+\mathbf{u}'.

\]

Using a diagonal approximation to the momentum operator gives, for a face flux,

\[

\dot m_f'=d_f(p_P'-p_N'),

\]

where df includes density, face area, and the inverse of an appropriate momentum coefficient. Substitution into continuity yields a pressure-correction equation of the form

\[

a_P^p p_P'=\sum_N a_N^p p_N'+b_P^p,

\]

where the source term is the mass imbalance of the provisional velocity field. Once p' is obtained, pressure, face fluxes, and cell velocities are corrected. The use of corrected face fluxes is essential: correcting cell-centre velocities alone does not guarantee discrete mass conservation.

7.5 The SIMPLE family

SIMPLE

The Semi-Implicit Method for Pressure-Linked Equations proceeds through the following cycle:

1. Guess or retain a pressure field.

2. Solve the discretized momentum equations using that pressure.

3. Construct a pressure-correction equation from continuity and approximate momentum relations.

4. Solve for pressure correction.

5. Correct pressure, face mass fluxes, and velocities.

6. Under-relax selected variables and repeat until residuals and conservation errors are acceptable.

Pressure under-relaxation is commonly written as

\[

p^{new}=p^{old}+\alpha_p p',

\]

where 0<αp≤1. Momentum under-relaxation modifies the diagonal coefficient and limits changes in velocity. Under-relaxation improves robustness for strongly nonlinear steady problems but slows convergence when chosen unnecessarily small.

The original SIMPLE approximation neglects some velocity-correction terms. It is robust but may converge slowly, especially on fine meshes or for strongly coupled flows.

SIMPLEC and PISO

SIMPLEC modifies the velocity-correction approximation so that the pressure correction has a stronger relationship with the actual velocity correction. It often permits larger relaxation factors and faster steady convergence.

The Pressure-Implicit with Splitting of Operators (PISO) algorithm performs multiple pressure-correction stages within one time step. After the first correction, the momentum equations are not fully re-solved; instead, additional corrections account for the effect of the first pressure correction on fluxes and subsequent continuity errors. PISO is attractive for transient calculations because it reduces the need for strong under-relaxation and can achieve good temporal coupling with one momentum solution per time step, subject to stability and accuracy requirements.

PIMPLE-type procedures combine outer SIMPLE-like iterations with inner PISO corrections. They are useful when a transient problem has strong nonlinearities, moving meshes, or large time steps requiring more than one correction cycle.

7.6 Projection methods

Projection methods are especially common for transient incompressible flow. A typical fractional-step procedure first computes an intermediate velocity without the new pressure:

\[

\frac{\mathbf{u}^-\mathbf{u}^n}{\Delta t}= -\nabla\cdot(\mathbf{u}\mathbf{u})^n+\nu\nabla^2\mathbf{u}^+\mathbf{f}.

\]

The velocity is then projected onto a divergence-free field:

\[

\mathbf{u}^{n+1}=\mathbf{u}^*-\frac{\Delta t}{\rho}\nabla p^{n+1}.

\]

Applying divergence and enforcing continuity gives the pressure Poisson equation

\[

\nabla^2p^{n+1}=\frac{\rho}{\Delta t}\nabla\cdot\mathbf{u}^*.

\]

Projection methods have a clear physical and mathematical interpretation: advance the momentum and then remove the irrotational component that violates incompressibility. Their accuracy depends on the splitting treatment, pressure boundary conditions, and spatial discretization. Basic pressure-correction projection may produce boundary-layer errors near no-slip walls; incremental and rotational forms improve accuracy in many situations.

A pressure Poisson equation is elliptic, so its solution communicates the incompressibility constraint globally. Efficient multigrid or Krylov solvers are therefore central to the performance of projection-based CFD.

7.7 Fully coupled algorithms

In a coupled approach, pressure and velocity unknowns are assembled into one block system, schematically

\[

\begin{bmatrix}

A_u & G\\

D & 0

\end{bmatrix}

\begin{bmatrix}

\mathbf{u}\\p

\end{bmatrix}

=

\begin{bmatrix}

\mathbf{b}_u\\\mathbf{b}_p

\end{bmatrix}.

\]

Here G is the pressure-gradient operator and D is the divergence operator. The zero pressure block reflects the absence of an independent pressure evolution equation. The system is a saddle-point problem and may be indefinite.

Coupled solvers can converge rapidly when pressure and velocity are strongly linked, for example in low-Mach-number flows, rotating flows, or problems with severe pressure gradients. They reduce splitting error and may require less under-relaxation. Their costs include larger memory demand, more complex preconditioning, and greater sensitivity to linear-solver quality.

Block preconditioners, approximate Schur complements, algebraic multigrid, and flexible Krylov methods are commonly used. A coupled method is not automatically superior: a well-designed segregated method may be more economical for large industrial meshes, especially when the physics is only moderately coupled.

7.8 A practical algorithm-selection framework

For steady internal flows on general meshes, SIMPLE or SIMPLEC remains a dependable choice. PISO is often advantageous for transient flows where accurate time advancement and strong pressure–velocity coupling are required. Projection methods are natural for finite-difference, finite-volume, and spectral solvers designed around time-dependent incompressible equations. Fully coupled methods become attractive when splitting severely limits convergence or when several fields are tightly interdependent.

The selection should consider:

• whether the problem is steady or transient;
• the expected Courant number and time-step size;
• mesh quality and grid topology;
• strength of convection, rotation, buoyancy, and source terms;
• memory available for block systems;
• the quality of pressure and momentum preconditioners;
• whether exact mass conservation at faces is required;
• the cost of nonlinear iterations relative to linear solves.

Algorithm selection cannot compensate for poor boundary conditions, inadequate mesh resolution, or inconsistent discretization. Solver choice is part of a complete numerical design rather than an isolated setting.

7.9 A detailed illustrative case: pressure-driven flow through a sudden expansion

Background

Consider an incompressible fluid entering a circular pipe, passing through a sudden expansion, and leaving through a longer downstream section. The objective is to predict the recirculation zone, pressure recovery, and permanent pressure loss.

Situation and problem formulation

The inlet flow rate is known from a pump specification, while the downstream static pressure is known relative to a reference. No-slip conditions apply on the pipe walls. The expansion creates separation and a recirculation region, so an outlet placed too close to the expansion may experience reversed flow.

A suitable initial field is a low-speed or fully developed inlet profile extended through the domain, followed by a pressure projection or correction. At the inlet, the velocity profile and turbulence quantities are specified consistently with the flow rate. At the outlet, static pressure is prescribed, while velocity gradients are treated with an outlet condition that can accommodate limited reverse flow. The outlet is placed sufficiently far downstream that the separated shear layer has largely redeveloped.

Analysis and algorithm

A collocated finite-volume grid is selected because the expansion is represented conveniently on an unstructured mesh. Rhie–Chow-type momentum interpolation is used for face mass fluxes. The momentum equations are solved with second-order spatial discretization after an initial first-order stabilization phase. SIMPLEC is used for the steady solution, with moderate momentum relaxation and pressure relaxation adjusted according to continuity convergence.

The provisional momentum solution generally produces a mass imbalance in cells near the expansion. The pressure-correction equation redistributes face fluxes: it reduces flux entering cells where the provisional divergence is positive and increases it where the provisional divergence is negative. Corrected face fluxes, rather than merely corrected cell velocities, are used in the next iteration.

Because separation is sensitive to mesh resolution, local refinement is applied near the expansion corner and in the shear layer. Grid studies compare pressure recovery and reattachment length. If the flow is inherently unsteady, forcing a steady solution may yield oscillatory residuals or a misleading averaged recirculation region; in that case a transient PISO calculation is more appropriate.

Outcome

A credible solution should exhibit a near-zero integrated continuity error, a physically plausible recirculation zone, and pressure recovery that becomes insensitive to further outlet extension and mesh refinement. The computed pressure drop should be checked against a control-volume momentum balance and, where available, experimental or validated reference data.

Learning points

The case illustrates four important principles. First, outlet placement is part of boundary-condition design. Second, separated flow makes simple zero-gradient outlet assumptions less reliable. Third, collocated grids require pressure–flux stabilization. Fourth, convergence of residuals alone is insufficient; conservation, integral quantities, and grid sensitivity must also be examined.

7.10 Common failure modes and limitations

A checkerboard pressure pattern usually indicates an inconsistent collocated pressure–velocity interpolation or an incorrectly assembled pressure equation. It should not be hidden by excessive numerical diffusion.

Oscillating residuals may result from excessive relaxation, a time step that is too large, a physically unsteady flow forced into a steady formulation, poor outlet treatment, or an ill-conditioned pressure system. Reducing the relaxation factor may help, but it can also conceal an underlying formulation error.

A globally converged residual does not guarantee local mass conservation if the pressure equation, boundary fluxes, or correction fluxes are inconsistent. Always monitor the maximum cell continuity error and integrated inlet–outlet imbalance.

Specifying both velocity and pressure at an inlet, prescribing incompatible mass flow rates, fixing pressure at multiple unrelated locations, or imposing a wall pressure without physical justification are common errors. Another frequent mistake is applying a symmetry boundary to a flow that contains asymmetric instability.

Pressure-correction methods use approximations to the inverse momentum operator. These approximations may become weak for highly skewed grids, strong nonlinearity, large time steps, or strongly anisotropic diffusion. Coupled algorithms reduce some splitting effects but do not eliminate discretization error or poor boundary formulation.

7.11 Professional practice, verification, and emerging developments

In professional CFD, boundary conditions should be traceable to measured operating data, design specifications, or defensible physical assumptions. The analyst should document reference pressure, mass-flow conventions, turbulence quantities, wall treatment, outlet distance, and the criteria used to declare convergence.

Verification includes checking code implementation, discrete conservation, manufactured solutions, pressure-gradient recovery, and grid or time-step convergence. Validation compares predictions with experiments or trusted reference solutions. A converged calculation can still be invalid if the physical model or boundary data are inappropriate.

Modern solvers increasingly use geometric and algebraic multigrid, GPU acceleration, matrix-free operators, adaptive time stepping, and block preconditioning. Automated mesh adaptation can refine regions of high vorticity, pressure gradient, or estimated discretization error. Data-driven methods and machine learning are being explored for relaxation prediction, reduced-order pressure correction, surrogate modelling, and error estimation. These tools should support—not replace—conservation checks, physical reasoning, and numerical verification.

For moving boundaries and fluid–structure interaction, pressure–velocity coupling is extended to include mesh motion and structural forces. Partitioned algorithms may require subiterations to avoid added-mass instability, while monolithic formulations solve fluid and structure variables together. Similar coupling issues arise in multiphase flow, porous media, combustion, and low-Mach reacting systems.

Best Practices

1. Define the physical problem before selecting numerical boundary types.

2. Use one pressure reference in an incompressible domain and verify global flux compatibility.

3. Distinguish prescribed velocity, prescribed pressure, and flux conditions; do not over-specify a boundary.

4. Use staggered storage or a proven momentum-interpolation method on collocated grids.

5. Correct face mass fluxes consistently with the pressure correction.

6. Monitor continuity imbalance, momentum residuals, integral quantities, and solution changes.

7. Place outlets away from separation and strong recirculation whenever possible.

8. Begin difficult steady calculations with robust discretization and gradually increase accuracy.

9. Select SIMPLE-family, PISO, projection, or coupled methods according to the temporal and physical character of the problem.

10. Verify mesh, time-step, and boundary-location independence before using results for design decisions.

Chapter Summary

In incompressible CFD, pressure is a constraint variable whose principal role is to enforce discrete continuity. Since velocity depends on pressure gradients and pressure depends on velocity fluxes, pressure–velocity coupling is central to both mathematical correctness and numerical stability. Initial and boundary conditions must be physically compatible, globally conservative, and matched to the actual flow configuration.

Staggered grids directly connect pressure differences to face velocities, whereas collocated grids require momentum-consistent face interpolation to prevent checkerboard pressure fields. SIMPLE and SIMPLEC are robust segregated methods for steady problems; PISO and projection methods are well suited to transient flow; and fully coupled block methods can be advantageous for strongly linked systems. No algorithm can compensate for incompatible boundary data or inadequate verification. Reliable CFD therefore combines appropriate formulation, consistent discretization, efficient linear algebra, and disciplined physical and numerical assessment.

Key Terms

Pressure–velocity coupling: Numerical treatment that enforces the mutual dependence of pressure and velocity in incompressible flow.

Pressure correction: An auxiliary pressure variable used to reduce mass imbalance and update pressure and fluxes.

Pressure reference: A specified pressure value that removes the arbitrary additive constant in incompressible pressure.

Staggered grid: Arrangement in which pressure and velocity components are stored at different geometric locations.

Collocated grid: Arrangement in which pressure and all velocity components are stored at cell centres.

Checkerboard pressure: An oscillatory, nonphysical pressure pattern caused by inadequate pressure–velocity coupling.

Momentum interpolation: A face-velocity or face-flux reconstruction designed to maintain pressure–velocity coupling on collocated grids.

SIMPLE: Semi-Implicit Method for Pressure-Linked Equations, a segregated pressure-correction algorithm.

SIMPLEC: A modified SIMPLE method with an improved velocity-correction approximation.

PISO: Pressure-Implicit with Splitting of Operators, using multiple pressure corrections within a time step.

Projection method: Transient method that projects an intermediate velocity onto a divergence-free field.

Pressure Poisson equation: Elliptic equation obtained by applying continuity to a pressure-based velocity correction.

Under-relaxation: Limitation of an iterative variable update to improve nonlinear stability.

Face mass flux: Mass flow rate crossing a control-volume face.

Continuity residual: Measure of the discrete mass imbalance in a control volume or whole domain.

Pressure outlet: Boundary condition that specifies pressure while allowing velocity to adjust from the interior solution.

Convective outlet: Outlet condition that approximates downstream transport of disturbances.

Saddle-point system: Indefinite coupled algebraic system characteristic of incompressible velocity–pressure formulations.

Schur complement: Reduced operator obtained by eliminating one block of a coupled linear system.

Divergence-free field: Velocity field whose discrete divergence is zero in every control volume.

Review Questions

Short Answer Questions

1. Why is pressure not obtained from an independent evolution equation in incompressible flow?

2. What is the purpose of specifying a pressure reference?

3. State the main difference between staggered and collocated variable arrangements.

4. What is the function of a pressure-correction equation?

5. Why must corrected face mass fluxes be used in a finite-volume pressure-correction method?

Long Answer Questions

1. Explain the formulation of initial and boundary conditions for an incompressible internal-flow problem, including inlet, outlet, wall, symmetry, and pressure-reference conditions.

2. Describe the origin of checkerboard pressure fields on collocated grids and explain how momentum interpolation prevents them.

3. Derive the conceptual sequence of the SIMPLE algorithm and discuss the role of under-relaxation.

4. Compare SIMPLE, SIMPLEC, PISO, projection, and fully coupled algorithms in terms of formulation, convergence, and application.

5. Explain why outlet boundary conditions are particularly difficult in separated and recirculating flows.

Analytical / Application Questions

1. A simulation specifies a fixed inlet velocity, a fixed outlet velocity, and fixed pressure at both inlet and outlet. Assess whether the problem is over-specified and propose a consistent alternative.

2. A collocated-grid channel calculation converges in momentum residuals but develops alternating high and low pressure values between adjacent cells. Diagnose the cause and recommend corrective actions.

3. In a sudden-expansion calculation, the outlet is close to the recirculation region and the mass imbalance oscillates despite reducing relaxation. Design a revised numerical strategy involving boundary placement, algorithm selection, and verification measures.

22. Multiple Choice Questions with Answers

Question 1

Why must a pressure reference be specified in a closed incompressible domain?

A. Pressure has no effect on velocity gradients

B. The pressure field is determined only up to an arbitrary additive constant

C. The density otherwise becomes compressible

D. It eliminates the need to solve continuity

Correct Answer: B. The pressure field is determined only up to an arbitrary additive constant

Question 2

Which arrangement naturally places a normal velocity component between the two pressure locations used to calculate its pressure gradient?

A. Collocated arrangement

B. Random arrangement

C. Staggered arrangement

D. Modal arrangement

Correct Answer: C. Staggered arrangement

Question 3

What is the principal purpose of Rhie–Chow-type momentum interpolation?

A. To enforce the energy equation

B. To prevent pressure checkerboarding on collocated grids

C. To impose a no-slip wall condition

D. To calculate fluid density from temperature

Correct Answer: B. To prevent pressure checkerboarding on collocated grids

Question 4

In SIMPLE, the source term of the pressure-correction equation is primarily related to:

A. The provisional mass imbalance

B. The molecular weight of the fluid

C. The wall roughness alone

D. The absolute pressure reference

Correct Answer: A. The provisional mass imbalance

Question 5

Which method performs multiple pressure corrections within a single time step?

A. SIMPLE

B. PISO

C. Gauss elimination only

D. Explicit Euler without projection

Correct Answer: B. PISO

Question 6

For a prescribed impermeable wall velocity, the pressure-correction boundary condition commonly follows from:

A. The requirement that the corrected normal velocity satisfy impermeability

B. An arbitrary fixed pressure at the wall

C. The inlet mass-flow rate only

D. The equation of state

Correct Answer: A. The requirement that the corrected normal velocity satisfy impermeability

Question 7

A pressure-driven channel calculation should generally prescribe:

A. Both inlet velocity and inlet pressure independently

B. A pressure difference or gradient, with wall conditions and a pressure reference

C. Velocity at every boundary including solid walls

D. Zero pressure everywhere

Correct Answer: B. A pressure difference or gradient, with wall conditions and a pressure reference

Question 8

What is the defining operation of a projection method?

A. Projection of an intermediate velocity onto a divergence-free space

B. Projection of pressure onto the wall

C. Projection of density onto the energy equation

D. Projection of the mesh onto the inlet

Correct Answer: A. Projection of an intermediate velocity onto a divergence-free space

Question 9

Why can a zero-gradient velocity outlet be unreliable when strong recirculation reaches the boundary?

A. It always violates the equation of state

B. The flow is no longer predominantly leaving the domain, so the assumed downstream behaviour is inappropriate

C. It forces pressure to be discontinuous everywhere

D. It changes a staggered grid into a collocated grid

Correct Answer: B. The flow is no longer predominantly leaving the domain, so the assumed downstream behaviour is inappropriate

Question 10

Which situation most strongly motivates consideration of a fully coupled velocity–pressure solver?

A. A weakly coupled, well-resolved laminar problem with abundant splitting convergence

B. A problem with strong pressure–velocity interaction where segregated splitting severely limits convergence

C. A calculation with no pressure variable

D. A purely post-processing operation

Correct Answer: B. A problem with strong pressure–velocity interaction where segregated splitting severely limits convergence