The movement of fluid in the physical domain is driven by various properties. For the purpose of bringing the behavior of fluid flow to light and developing a mathematical model, those properties have to be defined precisely as to provide a transition between the physical and the numerical domain. Velocity, pressure, temperature, density, and viscosity are the main properties that should be considered simultaneously when conducting a fluid flow examination. In accordance with the physical phenomena such as combustion, multiphase flow, turbulence, mass transport, etc., those properties diversify enormously and can be categorized into kinematic, transport, thermodynamic, and other miscellaneous properties\(^1\).

Thermo-fluid incidents directed by governing equations are based on the laws of conservation. The Navier-Stokes (N-S) equations is the broadly applied mathematical model to examine changes on those properties during dynamic and/or thermal interactions. The equations are adjustable regarding the content of the problem and are expressed based on the principles of conservation of mass, momentum, and energy\(^1\):

Conservation of Mass: Continuity Equation

Conservation of Momentum: Newton’s Second Law

Conservation of Energy: First Law of Thermodynamics or Energy Equation

Although some sources specify the expression of Navier-Stokes equations merely for the conservation of momentum, some of them also use all equations of conservation of the physical properties. Regarding the flow conditions, the N-S equations are rearranged to provide affirmative solutions in which the complexity of the problem either increases or decreases. For instance, having a numerical case of turbulence according to the pre-calculated Reynolds number requires an appropriate turbulent model to be applied to obtain credible results.

Despite the fact that the motion of fluid is an exploratory topic for human beings, the evolution of mathematical models emerged at the end of 19\(^{th}\) century after the industrial revolution. The initial appropriate description of the viscous fluid motion had been indicated in the paper “Principia” by Sir Isaac Newton (1687) in which dynamic behavior of fluids under constant viscosity was investigated\(^1\). Later, Daniel Bernoulli (1738) and Leonhard Euler (1755) subsequently derived the equation of inviscid flow which is now expressed as Euler’s inviscid equations. Even though Claude-Louis Navier (1827), Augustin-Louis Cauchy (1828), Siméon Denis Poisson (1829), and Adhémar St.Venant (1843) had carried out studies to explore the mathematical model of fluid flow, they had overlooked the viscous (frictional) force. In 1845, Sir George Stokes had derived the equation of motion of a viscous flow by adding Newtonian viscous terms, thereby the Navier-Stokes Equations had been brought to their final form which has been used to generate numerical solutions for fluid flow ever since\(^{1,2}\).

The observation method of fluid flow based on kinematic properties is a fundamental issue for generating a convenient mathematical model. Movement of fluid can be investigated with either Lagrangian or Eulerian methods\(^3\). Lagrangian description of fluid motion is based on monitoring a fluid particle that is large enough to detect properties. Between the initial coordinates at time \(t_0\) and coordinates of the same particle at time \(t_1\), millions of separate particles have to be examined through the path that is almost impossible to follow. In the Eulerian method, any specific particle across the path is not followed; instead, the velocity field as a function of time and position is examined. The missile example (Figure 3) precisely fits to emphasize these methods.

The Lagrangian formulation of motion is always time-dependent. As \(a\), \(b\), and \(c\) are the initial coordinates of a particle; \(x\), \(y\), and \(z\) are coordinates of the same particle at time \(t\). Description of motion for Lagrangian:

In the Eulerian method, \(u\), \(v\) and \(w\) are the components of velocity at the point \((x,y,z)\) while \(t\) is the time. The velocity components \(u\), \(v\) and \(w\) are the unknowns which are functions of the independent variables \(x\), \(y\), \(z\) and \(t\). The description of motion with the Eulerian method for any particular value of \(t\) is:

The equations of conservation in the Eulerian system in which fluid motion is described are expressed as Continuity Equation for mass, Navier-Stokes Equations for momentum and Energy Equation for the first law of Thermodynamics. The equations are all considered simultaneously to examine fluid and flow fields.

Conservation of Mass

The mass in the control volume can be neither be created nor destroyed. The conservation of mass states that the mass flow difference throughout the system between inlet and outlet is zero:

The momentum in a control volume is kept constant, which implies the conservation of momentum that we call ‘The Navier-Stokes Equations’. The description is set up in accordance with the expression of Newton’s Second Law of Motion:

$$ F=m\cdot a \tag6$$

where \(F\) is the net force applied to any particle, \(a\) is the acceleration, and \(m\) is the mass. In case of a fluid, it is convenient to express the equation in terms of the volume of the particle as follows:

in which \(f\) is the force exerted on the fluid particle per unit volume, and \(f_{body}\) is the applied force on the whole mass of fluid particles as below:

$$ f_{body}=p\cdot g \tag8$$

where \(g\) is the gravitational acceleration. External forces which are deployed through the surface of fluid particles, \(f_{surface}\) is expressed through pressure and viscous forces as:

where \(\tau_{ij}\) is stress tensor. According to the general deformation law of Newtonian viscous fluid given by Stokes, \(\tau_{ij}\) is expressed as\(^2\):

$$ \tau_{ij}=-p\delta_{ij}+\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)+\delta_{ij}\lambda\nabla\cdot V \tag{10}$$

Hence, Newton’s equation of motion can be specified in the form as follows:

Substitution of equation (10) into (11) results in the Navier-Stokes equations of Newtonian viscous fluid in one equation:

$$ \underbrace{\rho\frac{DV}{Dt}}_{I} = \underbrace{\rho\cdot g} _ {II} – \underbrace{\nabla p} _ {III}+\underbrace{\frac{\partial}{\partial x _ i}\left[\mu\left(\frac{\partial v _ i}{\partial x _ j} + \frac{\partial v _ j}{\partial x _ i}\right)+\delta _ {ij} \lambda\nabla\cdot V\right]} \tag{12} _ {IV}$$

\(I\): Momentum convection \(II\): Mass force \(III\): Surface force \(IV\): Viscous force

where static pressure is \(p\) and gravitational force is \(\rho\vec{g}\). Equation (12) is convenient for fluid and flow fields which are both transient and compressible. \(D/Dt\) indicates the substantial derivative as follows:

If the density of the fluid is constant, the equations are greatly simplified in which the viscosity coefficient \(\mu\) is assumed constant and \(\nabla\cdot V=0\) in equation (12). Thus, the Navier-Stokes equations for an incompressible three-dimensional flow can be expressed as follows:

$$ \rho\frac{DV}{Dt}=\rho g-\nabla p + \mu\nabla^2V \tag{14}$$

For each dimension when the velocity is \(V(u,v,w)\):

\(p\), \(u\), \(v\) and \(w\) are unknowns where a solution is sought by application of both continuity equation and boundary conditions. Besides, the energy equation has to be considered if any thermal interaction is available in the problem.

Conservation of Energy

Conservation of Energy is the first law of thermodynamics which states that the sum of the work and heat added to the system will result in the increase of the total energy of the system:

$$ dE_t=dQ+dW \tag{18}$$

where \(dQ\) is the heat added to the system, \(dW\) is the work done on the system, and \(dE_t\) is the increment in the total energy of the system. One of the common types of energy equation is:

\(I\): Local change with time \(II\): Convective term \(III\): Pressure work \(IV\): Heat flux \(V\): Heat dissipation term

The Navier-Stokes equations have a non-linear structure with various complexities and thus it is hardly possible to conduct an exact solution for those equations. Consequently, different assumptions are required to grind the equations to a possible solution.

The mathematical model merely gives ties among parameters that are part of the whole process. Hence, the solution of the Navier-Stokes equations can be realized with either analytical or numerical methods. The analytical method only compensates solutions in which non-linear and complex structures in the Navier-Stokes equations are ignored within several assumptions. It is only valid for simple/fundamental cases such as Couette flow, Poisellie flow, etc\(^3\). On the other hand, almost every case in fluid dynamics comprises non-linear and complex structures in the mathematical model which cannot be ignored. Hence, the solutions of the Navier-Stokes equations are carried out within several numerical methods, the omnipresence of Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). A step by step computational analysis of fluid flow can be described as shown in Figure 4.

Time Domain

The analysis of fluid flow can be conducted in either steady (time-independent) or unsteady (time-dependent) conditions. In case the flow is steady, it means the motion of fluid and parameters do not rely on the change in time, the term \(\frac{\partial()}{\partial t}=0\) where the continuity and momentum equations are re-derived as follows:

While the steady flow assumption negates the effect of some non-linear terms and provides a convenient solution, the variation of density is still a hurdle that keeps the equation in a complex formation.

Compressibility

Due to the malleable structure of fluids, the compressibility of particles is a significant issue. Despite the fact that all types of fluid flow are compressible in a various range of molecular structure, most of them can be assumed incompressible in which the density changes are negligible. Thus, the term \(\frac{\partial\rho}{\partial t}=0\) is thrown away regardless of whether the flow is steady or not, as below:

As the incompressible flow assumption provides reasonable equations, the application of steady flow assumption concurrently enables us to ignore non-linear terms where \(\frac{\partial()}{\partial t}=0\). Having said that, high speed flows where the velocity is beyond a critical limit cannot be assumed incompressible. “The Mach Number” is a dimensionless number that is convenient to investigate such fluid flows\(^3\):

$$ Ma=\frac{V}{a}\ \tag{24}$$

where \(Ma\) is the Mach number, \(V\) is the velocity of flow, and \(a\) is the speed of sound at \(340.29 \frac{m}{s}\) at sea level. When the Mach number is lower than 0.3, the assumption of incompressibility is acceptable. On the contrary, the change in density cannot be negligible where \(Ma>0.3\) – high speed flows. For instance, if the velocity of a car is higher than 100\(\frac{m}{s}\), the suitable approach to conduct credible numerical analysis is the compressible flow. Apart from velocity, the effect of thermal properties on the density changes has to be considered in geophysical flows\(^3\).

Low and High Reynolds Number

The Reynolds number, the ratio of inertial and viscous effects, is also effective on Navier-Stokes equations to truncate the mathematical model. While \(Re\longrightarrow\infty\), the viscous effects are presumed negligible and the viscous terms in Navier-Stokes equations are thrown away. The simplified form of Navier-Stokes equation, described as Euler equation, can be specified as follows\(^8\):

Even though viscous effects are relatively important for fluids, the inviscid flow model partially provides a reliable mathematical model to predict real process for some specific cases. For instance, high-speed external flow over bodies is a broadly used approximation where the inviscid approach reasonably fits. While \(Re\ll1\), the inertial effects are assumed negligible where related terms in Navier-Stokes equations drop out. The simplified form of Navier-Stokes equations is called either creeping flow or Stokes flow\(^8\):

Having tangible viscous effects, creeping flow is a suitable approach to investigate the flow of lava, swimming of microorganisms, flow of polymers, lubrication, etc.

Turbulence

The behavior of the fluid under dynamic conditions can be classified as laminar and turbulent. The laminar flow is orderly in which the motion of a fluid can be predicted precisely. The turbulent flow has a chaotic behavior and therefore it is hard to predict the fluid flow which shows a chaotic behavior. The Reynolds number predicts the behavior of fluid flow whether laminar or turbulent regarding several properties such as velocity, length, viscosity, and also type of flow. Whilst the flow is turbulent, a proper mathematical model is selected to carry out numerical solutions. Various turbulent models are available in the literature and each of them has a slightly different structure to examine chaotic fluid flow.

Turbulent flow can be applied to the Navier-Stokes equations to model the chaotic behavior. Apart from the laminar, transport quantities of the turbulent flow, it is driven by instantaneous values. Direct numerical simulation (DNS) is the approach to solving the N-s equations with instantaneous values. Having distinct fluctuations varying in a broad range, DNS needs enormous effort and expensive computational facilities. To avoid those hurdles, the instantaneous quantities are reinstated by the sum of their mean and fluctuating parts as follows:

where \(u\), \(v\), and \(w\) are velocity components and \(T\) is temperature. The differences among values are shown in Figure 5 both for steady and unsteady conditions:

Instead of instantaneous values that cause non-linearity, carrying out a numerical solution with mean values provides an appropriate mathematical model which is named “The Reynolds-averaged Navier-Stokes (RANS) equation”\(^4\). The fluctuations can be negligible for most engineering cases. Thus, the RANS turbulence model is a procedure to close the system of mean flow equations. The general form of The Reynolds-averaged Navier-Stokes (RANS) equation can be specified as follows:

The turbulence model of RANS varies in regards to methods such as k-omega, k-epsilon, k-omega-SST, and Spalart-Allmaras. Likewise, Large Eddy Simulation (LES) is another mathematical method for turbulent flows which is also comprehensively applied for several cases. Robust LES ensures more accurate results than RANS but requires much more time and computer memory. As in DNS, LES resolves the larger eddies but models the smaller ones\(^4\).

The simple form of the Navier-Stokes equations only encompasses the change in properties such as velocity, pressure, and density under dynamic conditions for one phase laminar flow. Most engineering applications require further mathematical models for numerical simulation. Some common engineering problems and their relevant mathematical models are given as below:

Airflow in a duct: Single phase flow, laminar/turbulent, steady/unsteady

Water flow in an open channel: Multiphase flow, laminar/turbulent, steady/unsteady

Combustion in cylinder: Multiphase flow, laminar/turbulent, unsteady, chemical reaction, heat transfer, mass transfer

Condensation/evaporation of water: Multiphase flow, laminar/turbulent, unsteady, heat transfer, mass transfer

Microfluidics

The Navier-Stokes equations cannot compensate for the physical model of the flow at very small scales such as the motion of single bacteria — also called microfluidics. Thereby, it is convenient to either change or reinstate the Navier-Stokes model with a suitable mathematical model. The Knudsen number (Kn) is a dimensionless number that is the ratio of the mean free path of molecular structure to the observation scale. The preferred model in accordance with the Knudsen number is shown in Figure 6:

Real-time simulation

The Navier-Stokes equations have been considered crucial even in the animation/video game world\(^5\). For the purpose of generating a real-time animation, the application of these equations provides stunning results enhancing the reality\(^6\). They are the reason why modern video games appear to be realistic in many more ways than several years ago. Just think about the movement of flags in the wind: doesn’t this look absolutely realistic in modern games?!

Stokes, George (1851). “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums”. Transactions of the Cambridge Philosophical Society. 9: 8–106.

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