Read time: 10 minutes
Target audience: CFD Researchers/ Automobile Engineers/ Thermal-Fluid Industry/ Aero Industry
Written by: Dr. Tabish Wahidi

Background:

The Navier-Stokes (NS) equation is fundamental in describing fluid motion, but it present significant challenges and complexities when applied in computational fluid dynamics (CFD) simulations. These challenges arise from the mathematical nature of the equations, the physical phenomena they represent, and the computational methods used to solve them. Here’s a comprehensive explanation of these complexities, along with approaches to address them.

Introduction:

The NS equation is the cornerstone of CFD simulations, governing the motion of fluid substances like liquids and gases. Named after Claude-Louis Navier and George Gabriel Stokes, the equation describe how the velocity field of a fluid evolves over time and space under the influence of various forces. The NS equation can describe the motion of fluids, whether they are compressible or incompressible, Newtonian or non-Newtonian. Equation 1 below is the general form of NS equation, incorporates the conservation of momentum, accounts for variations in density and viscosity, and can be used to model a wide range of fluid behaviours.

Where:

1. 1. ρ is the fluid density
   2. u=(u ,v,w) is the velocity vector
   3. t is time.
   4. p is the pressure.
   5. τ is the stress tensor, which includes both viscous and other stress contributions.
   6. f is the body force per unit volume, such as gravity.
   7. Inertia term: ρ(∂u/∂t) : represent rate of change of momentum with time.
   8. Convective term: ρ(u.∇)u : This nonlinear term represents the transport of momentum by the fluid flow itself.
   9. Pressure Gradient term: -∇p : This term represents the force due to pressure differences within the fluid.
   10. Viscous term: (μ.∇^2 )u : This term accounts for the internal friction within the fluid due to its viscosity, leading to the diffusion of momentum. This term comes under stress tensor (τ)
   11. Body Force term: f : This term includes external forces acting on the fluid, such as gravity or electromagnetic forces. 

Complexities and its Solution Approach:

A. Nonlinearity:

The convection term (u.∇)u in the NS equation is the nonlinear term which leads to several issues:

1. 1. Chaotic Behaviour: Nonlinear systems can exhibit chaotic behaviour, where small changes in initial conditions lead to vastly different outcomes, making predictions difficult.
   2. Complex Solution Space: The nonlinearity complicates the mathematical structure of the equations, making analytical solutions possible only for simple cases and requiring numerical methods for practical problems.

Solution Approach:

1. 1. Iterative Methods: Nonlinear equations are typically solved using iterative methods like Newton-Raphson, which linearize the equations at each step. Careful convergence criteria and under-relaxation techniques can help manage the instability.
   2. Stabilization Techniques: Methods like artificial viscosity, upwinding, or flux limiters are used to stabilize numerical solutions by damping unphysical oscillations without overly smoothing the solution.

B. Turbulence

Turbulence, characterized by chaotic and irregular fluid motion, involves a wide range of spatial and temporal scales. The NS equations in turbulent flows are highly sensitive to initial conditions and boundary conditions, making them difficult to solve directly:

1. 1. Scale Range: Resolving all scales in a turbulent flow requires extremely fine computational grids and small-time steps, making direct numerical simulation (DNS) computationally prohibitive for most practical applications.
   2. Modeling Complexity: Simplified models like Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) are often used, but these models introduce uncertainties and require careful calibration.

  Solution Approach:

1. 1. Turbulence Models:
      1. RANS Models: These models average the NS equations over time, reducing the computational load but requiring turbulence closure models like k-ε or k-ω to represent the effects of turbulence. RANS models are widely used for steady-state simulations.
      2. LES Models: LES resolves large turbulent structures while modeling smaller scales, offering a balance between accuracy and computational cost. It is suitable for unsteady and more complex flows.
      3. Hybrid Models: Combining RANS and LES (e.g., Detached Eddy Simulation, or DES) leverages the strengths of both methods, using RANS in boundary layers and LES in regions of free turbulence.

C. Boundary Conditions

Applying appropriate boundary conditions is critical to the accuracy of CFD simulations. Improperly defined boundary conditions can lead to unphysical results or convergence issues:

1. 1. Complex Geometries: Real-world geometries are often complex, requiring detailed and accurate boundary representation.
   2. Inflow and Outflow Conditions: Defining conditions at the inlet and outlet of the simulation domain is challenging, particularly in open or unbounded flows, where artificial reflections or incorrect flow developments can occur.

   Solution Approach:

1. 1. Mesh Generation: Advanced mesh generation techniques, including adaptive mesh refinement (AMR) and body-fitted meshes, help accurately represent complex geometries. Structured and unstructured grids are used depending on the geometry’s complexity.
   2. Boundary Condition Techniques:
      1. Inflow/Outflow Conditions: Techniques like convective outflow or non-reflective boundary conditions minimize artificial reflections. Periodic boundary conditions can also be used in cases of repeating patterns.
      2. Wall Functions: In turbulent flows, wall functions approximate the behavior of the boundary layer near solid walls, reducing the need for excessively fine grids in those regions.

D. Numerical Stability and Convergence

Ensuring the numerical stability and convergence of solutions to the NS equations is a significant challenge:

1. 1. Time Stepping: Choosing an appropriate time step is crucial in transient simulations. Too large a time step can lead to instability, while too small a time step increases computational cost.
   2. Discretization Errors: Numerical methods introduce discretization errors, which can lead to inaccuracies if not properly managed.

  Solution Approach:

1. 1. CFL Condition: The Courant-Friedrichs-Lewy (CFL) condition guides the selection of time steps relative to the spatial resolution to maintain stability in explicit time-stepping methods.
   2. Implicit Methods: Implicit time-stepping methods, while more computationally expensive per step, allow for larger time steps and better stability, particularly in stiff problems.
   3. High-Order Schemes: High-order numerical schemes reduce truncation errors and improve accuracy. Adaptive methods adjust the resolution based on the solution’s behaviour, concentrating computational effort where it’s needed most.

E. Computational Cost

The computational cost of solving the NS equations for realistic problems can be prohibitive, especially for high-fidelity simulations involving turbulence, multiphase flows, or complex geometries:

1. 1. Resource Intensity: High-resolution simulations require significant computational resources, both in terms of processing power and memory.
   2. Long Simulation Times: High-fidelity simulations, such as LES or DNS, can take days or even weeks to complete, making them impractical for many applications.

Solution Approach:

1. 1. Parallel Computing: Utilizing parallel computing, including distributed memory (MPI) and shared memory (OpenMP) approaches, allows simulations to be run on high-performance computing (HPC) clusters, significantly reducing computation time.
   2. Reduced-Order Models: Techniques like Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD) create reduced-order models that capture the essential dynamics of the flow with fewer computational resources.
   3. Multi-Fidelity Approaches: Combining high-fidelity simulations with lower-fidelity models or experimental data can provide accurate results while managing computational costs.

F. Physical Modeling

Accurately capturing the physical processes within a fluid, such as heat transfer, phase change, or chemical reactions, adds complexity to the NS equations:

1. 1. Multiphase Flows: Simulating interactions between different phases (e.g., liquid-gas interfaces) requires additional modeling and computational resources.
   2. Combustion and Heat Transfer: Coupling the NS equations with additional equations for energy and species concentration is necessary for problems involving heat transfer or chemical reactions, but this increases the complexity and computational load.

Solution Approach:

1. 1. Coupled Solvers: CFD solvers that can handle coupled phenomena (e.g., reacting flows, multiphase flows) are essential. These solvers simultaneously solve the NS equations along with additional equations for energy, species, and phase change.
   2. Phase-Field Models: For multiphase flows, phase-field models like the Volume of Fluid (VOF) or Level-Set methods accurately track interfaces between phases.
   3. Detailed Chemistry Models: In combustion simulations, detailed chemical kinetics models coupled with the NS equations provide accurate predictions of combustion behaviour, though at a high computational cost. Reduced chemistry models or tabulated chemistry approaches can be used to balance accuracy and efficiency.

Conclusion:

The challenges and complexities of solving the Navier-Stokes equations in CFD simulations are significant, involving issues related to nonlinearity, turbulence, boundary conditions, numerical stability, computational cost, and physical modeling. Addressing these challenges requires a combination of advanced numerical methods, computational resources, and physical modeling techniques. Despite these difficulties, ongoing advances in CFD technology and computing power continue to improve the ability to simulate complex fluid flows with increasing accuracy and efficiency.