Conservation of Mass, Momentum & Energy

What Is It?

The governing equations of fluid flow are expressions of three fundamental conservation laws: conservation of mass (continuity), conservation of momentum (Navier–Stokes equations) and conservation of energy. Every CFD solver, regardless of its specific formulation, solves discretised versions of these equations. Understanding the physical meaning of each equation — what is conserved, how, and what terms represent — is essential for interpreting CFD results and for diagnosing problems.

Why It Matters

The conservation equations are not abstract mathematics — they are the physical laws that the flow must obey. A CFD result that violates mass conservation, that produces momentum without a source, or that creates energy from nothing is physically wrong, regardless of how good the contours look. Understanding the equations allows the engineer to check whether the solver is satisfying conservation, to interpret the balance of terms in a specific flow region, and to recognise when a result is physically implausible.

Every CFD result is built on conservation. If mass is not conserved — if the mass flow in does not equal the mass flow out — the solution is not physically valid, no matter how well converged the residuals appear.

Conservation of Mass (Continuity)

The continuity equation expresses conservation of mass: the rate of change of mass within a control volume equals the net mass flux into the volume. For a compressible flow, the density may vary with time and position. For an incompressible flow, the density is constant and the equation simplifies — the net volume flux into any control volume must be zero.

Continuity equation (compressible):

∂ρ/∂t  +  ∇·(ρu)  =  0

where:
ρ  =  fluid density
t  =  time
u  =  velocity vector
∇·  =  divergence operator

Incompressible simplification (ρ = constant):

∇·u  =  0

(divergence of velocity is zero — volume in equals volume out)

Conservation of Momentum (Navier–Stokes)

The momentum equations express Newton's second law applied to a fluid: the rate of change of momentum equals the sum of forces acting on the fluid. The forces include pressure gradients, viscous stresses and body forces (such as gravity). The Navier–Stokes equations are the momentum conservation equations for a Newtonian fluid — one where the viscous stress is proportional to the strain rate.

Navier–Stokes (momentum conservation, simplified):

ρ (∂u/∂t + u·∇u)  =  −∇p  +  ∇·τ  +  ρg

where:
ρ (∂u/∂t)       =  local acceleration (rate of change at a point)
ρ (u·∇u)        =  convective acceleration (change due to fluid motion)
−∇p             =  pressure gradient force
∇·τ             =  viscous force (divergence of viscous stress tensor)
ρg              =  body force (gravity, etc.)

For Newtonian fluid:  τ = μ (∇u + ∇uᵀ)  — viscous stress proportional to strain rate

Physical Meaning of Each Term

Conservation of Energy

The energy equation expresses conservation of energy: the rate of change of energy within a control volume equals the net energy flux plus work done and heat added. For compressible flows and thermal-fluid problems, the energy equation is essential — it couples temperature to the flow field. For isothermal incompressible flows, the energy equation may be omitted if temperature is not of engineering interest.

Energy equation (simplified, summarised):

∂(ρE)/∂t  +  ∇·(ρEu)  =  ∇·(k∇T)  +  ∇·(τ·u)  −  ∇·(pu)  +  S_E

where:
E          =  total energy (internal + kinetic)
k∇T        =  heat conduction (Fourier's law)
τ·u        =  viscous dissipation (work of viscous stresses)
pu         =  pressure work
S_E        =  energy sources (chemical reactions, radiation, etc.)

Key terms: convection, conduction, pressure work, viscous dissipation

Energy Equation Terms

• Convection — energy transported by fluid motion
• Conduction — heat diffusion through the fluid (Fourier's law)
• Pressure work — work done by pressure forces on the fluid
• Viscous dissipation — irreversible conversion of kinetic energy to heat through friction
• Sources — chemical reactions, radiation, joule heating

Mass Imbalance as a Numerical Check

One of the most important and simplest checks in CFD is mass conservation. The mass flow entering the domain should equal the mass flow leaving (for steady flow) or the rate of mass change within the domain (for transient flow). A mass imbalance — the difference between inlet and outlet mass flow — indicates a numerical problem: a leak in the domain, a poorly converged solution, or an inconsistency in the boundary conditions. Mass imbalance should be checked and should be negligible relative to the total mass flow for a credible solution.

CFD CHECK: Confirm that mass flow into and out of the domain is consistent with the intended physical problem. A mass imbalance indicates a numerical problem — a leak, poor convergence or boundary condition inconsistency.

When a Simpler Method May Be Better

For simple internal flows — straight pipes, gradual diffusers — analytical methods (Bernoulli, Darcy–Weisbach) may provide adequate answers without CFD. For preliminary sizing, hand calculations based on empirical correlations may be sufficient. CFD is justified when the geometry is complex, the flow features (separation, recirculation, secondary flow) cannot be captured by analytical methods, or when detailed field data (pressure distributions, velocity profiles) is needed. Starting with a simple method and using CFD where the complexity warrants it is good engineering practice.

Global Conservation as an Engineering Closure Check

Local residual convergence is not the same as global physical closure. Integrate mass, momentum and energy across the complete domain and compare the imbalance with the physical through-flow or load being predicted. For an internal-flow system, inlet and outlet momentum flux plus pressure forces should reconcile with wall reaction; for a thermal problem, enthalpy transport, wall heat transfer and volumetric sources should close the energy balance. These integral checks can expose sign errors, unintended leakage boundaries, incorrect reference frames and post-processing mistakes that residual plots do not reveal. When CFD feeds another discipline, the same conservation quantities provide a powerful hand-off check: the force or heat transferred downstream should be consistent with the global balance of the CFD solution.

Residuals indicate numerical iteration behaviour; integral balances demonstrate whether the final solution respects the physical conservation statement used for engineering.

Key Takeaways

• The governing equations are conservation of mass, momentum and energy
• Continuity ensures mass conservation; Navier–Stokes ensures momentum conservation
• The energy equation couples temperature to the flow for compressible and thermal problems
• Mass imbalance is a critical numerical check — inlet and outlet mass flow must balance
• Every CFD result is built on these conservation laws — violating them means the result is physically wrong