A Physical Insight into Computational Fluid Dynamics and Heat Transfer

Abstract

Mathematical equations that describe all physical processes are valid only under certain assumptions. One of them is the minimum scales used for the given description. In fact, this prohibits the use of derivatives in the mathematical models of the physical processes. This article represents a derivative-free approach for the mathematical modelling. The proposed approach for CFD and numerical heat transfer is based on the conservation and phenomenological laws, and physical constraints on the minimum problem-dependent spatial and temporal scales (for example, on the average free path of molecules and the average time of their collisions for gases). This leads to the derivative-free governing equations (the discontinuum approximation) that are very convenient for numerical simulation. The theoretical analysis of governing equations describing the fundamental conservation laws in the continuum and discontinuum approximations is given. The article demonstrates the derivative-free approach based on the correctly defined macroparameters (pressure, temperature, density, etc.) for the mathematical description of physical and chemical processes. This eliminates the finite-difference, finite-volume, finite-element or other approximations of the governing equations from the computational algorithms.

1. Introduction: Three Eras in CFD

Three eras can be distinguished in the history of fluid dynamics depending on the governing equations and methods of their solution: analytical era (from 17th century to first half of the 20th century), transitional era (from first half of the 20th century to present) and computational era (starts today) [1,2,3].

The main goal of scientists in the analytical era of fluid dynamics was the establishment and analysis of the governing equations, verification of their solutions using experimental data and application of the obtained results for solving engineering problems. Johann Bernoulli, Daniel Bernoulli, Leonhard Euler, Jean-Baptiste le Rond d’Alembert and their followers considered an abstract mathematical object (infinitesimal volume of fluid) for describing the fluid macroflows. In fact, a fictitious continuous medium (also called a continuum) continuously distributed in space and time instead of the real liquid or gas was used for solving macroproblems of the fluid dynamics by known analytical methods of 17th century. This incorrect approach makes it possible to use the differential and integral calculus not only in theoretical (mathematical) fluid dynamics, but also in all continuum mechanics. Since any fraction of the continuum is continuum, only the sufficient differentiable functions can be used to describe the continuum motion by the differential equations.

Modern science distinguishes three structural levels of matter, as follows:

(1) The microcosm is composed of molecules, atoms, elementary particles, i.e., space of extremely small microobjects from ${10}^{-18}$ up to ${10}^{-8}$ metres. The characteristic distance of studied physical phenomena are called scale, and various measures (from the Planck length up to the micrometre) are used as microscales. The molecular and atomic physics describe the microcosm phenomena (heat conduction, phase transitions, chemical reactions, etc.) in the scale range of ${10}^{-18}$–${10}^{-8}$ metres;

(2) The macrocosm is a space of objects comparable to humans. Metre, second, kilogram and degree Kelvin are used as macroscales of length, time, mass and temperature;

(3) The megacosm involves planets, stars, galaxies, metagalaxies, i.e., huge cosmic scales and speeds where the distance is measured in light years and the time of existence of cosmic objects is millions and billions of years.

The differential and integral calculus formed the basis of mathematics of the XVII century. The application of this calculus to solve engineering problems required non-physical assumptions. The main assumption is a continuity hypothesis, i.e., a fictitious continuous medium (continuum) continuously distributed in space and in time is used instead of the real liquid or gas. The continuum approximation ignores molecular structure of substances, but the continuum has physical properties of a real liquid or gas on the macroscale lengths. In spite of this theoretical inaccuracy, the continuum approximation coupled with phenomenological laws makes it possible to solve many applied problems. Namely, the continuum allows us to constrain the volume of a liquid or gas to an interior point to obtain partial derivative equations that contradict the macrocosm laws.

The continuity hypothesis allows for using the partial differential equations to describe the properties and fluid flows. It was a new type of differential equations, and Euler described some of their types and also formulated methods for their integration. In 1755, Euler formulated the equations of an inviscid incompressible fluid flows (so called «Euler equations»)

$$\begin{array}{cc}\hfill \nabla \overrightarrow{\mathit{V}}& =0,\hfill \\ \hfill \rho {\displaystyle \frac{d\overrightarrow{\mathit{V}}}{dt}}& =\nabla p,\hfill \end{array}$$

where $\rho =const$ is the density of liquid or gas. These equations expresses the conservation of mass and momentum.

In 1822, Claude Louis Marie Henri Navier formulated the equations of a viscous incompressible fluid flows. Navier introduced an additional term accounting the viscosity of liquid or gas to the momentum equations. In 1829, Siméon Denis Poisson, using model concepts of molecular forces, formulated viscous compressible fluid flows. Later, a phenomenological derivation of these equations was given by Adhémar Jean Claude Barré de Saint-Venant and George Gabriel Stokes:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \overrightarrow{\mathit{V}}\right)& =0,\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \left(\rho \overrightarrow{\mathit{V}}\right)}{\partial t}}+(\rho \overrightarrow{\mathit{V}}·\nabla )\overrightarrow{\mathit{V}}& =-\nabla p+\eta △\overrightarrow{\mathit{V}}+\left(\zeta +{\displaystyle \frac{\eta }{3}}\right)\nabla div\overrightarrow{\mathit{V}},\hfill \end{array}$$

(1b)

where t is time, ∇ is the nabla operator, $△$ is the vector Laplace operator, $\eta$ is the dynamic viscosity coefficient, $\zeta$ is the second viscosity, $\rho$ is the density, p is the pressure, and $\overrightarrow{\mathit{V}}=(v^1,\dots ,v^n)$ is the velocity vector. The unknown p and $\overrightarrow{\mathit{V}}$ are the functions of time t and coordinate $x\in \Omega$, where $\Omega \subset {\mathbb{R}}^n$, $n=2,3$ is a domain where the fluid or gas is moving.

The differential and integral calculus defines the differential form of the governing equations in spite of rough physical assumptions. The nonlinear Navier–Stokes equations can be solved analytically in the simplest cases, so some semi-empirical theories such as the boundary layer approach have been proposed and developed to simplify mathematical description of the fluid flows [4,5].

The transitional era of fluid dynamics is based on the solver replacement. Approximate numerical methods for solving the (initial-)boundary value problems have replaced the differential and integral calculus. Computational physics is a branch of theoretical physics that uses numerical analysis and data structures to analyse and solve problems that involve various physical and chemical processes [6,7].

The current computational fluid dynamics (CFD) focuses on the more accurate numerical solution of governing partial differential equations (PDEs) [8]. For the given purpose, the (initial-)boundary value problems are replaced by their discrete analogues. The current target of computational mathematics is to obtain an sufficiently accurate approximation to the exact solution of the governing PDEs as the mesh size and time step approach to zero. This contradicts theoretical physics. The individual molecules have the standard physical microparameters: mass, momentum and energy. The standard physical macroparameters are defined by averaged microparameters of the individual molecules in a finite volume. For example, the density of a gas is the sum of the mass of the molecules divided by the finite volume which the gas occupies. The pressure and temperature of a gas are measure of the mean linear momentum and the mean kinetic energy of the molecules in the finite volume, respectively. It is clear that the averaging of the microparameters is correct only if this finite volume contains a sufficiently large number of molecules.

It is necessary to clarify the contradiction between the computational mathematics and the computational physics. Let the dimensions of the finite volumes be such that the gas or liquid contained in it can be considered as a continuous medium. The geometric scale is the representative size ℏ of the least volume, i.e., smaller scale processes cannot be distinguished. Therefore, it not possible to use classic definition of the derivative

$$g^{\prime }\left(a\right)=\underset{h\to 0}{lim}{\displaystyle \frac{g(a+h)-g\left(a\right)}{h}}$$

in point a for description of the physical processes because $\hslash \gg h$. On the other hand, the temperature, pressure or density of a gas at the point a cannot be defined according to classical physical theories. The derivatives and the macroparameters (density, pressure, temperature, etc.) cannot be correctly defined simultaneously [9].

The computational era is based on a dichotomy between the correct definition of derivatives and macroparameters. Since modern mathematical modelling does not imply the analytical solution of governing differential equations, derivative-free models with correctly defined macroparameters should be employed for the numerical investigation of physical phenomena. In this case, the mathematical model actually coincides with the finite volume-based scheme [10,11,12].

The goal of this article is to demonstrate the influence of correctly defined macroparameters on a mathematical description of the physical and chemical processes for computer-based solvers. The results obtained will be useful for black-box software used for numerical simulation of various physical and chemical processes.

2. The Discontinuum Approximation of Conservation Laws

First, let us consider the phenomenological derivation of the continuity equation, which expresses the fundamental law of mass conservation in the continuum approximation. Density is the most important parameter for describing flows of a compressible liquid or gas. Next, we will discuss and analyse the disadvantages of the classical continuity equation.

To illustrate the phenomenological technique, an arbitrary finite volume V in a continuous medium can be used [13,14]. Since the resulting equations depend on the volume shape, we will choose a rectangular parallelepiped as the volume V for clarity:

$$V=\left\{(t,x,y,z)\right|0⩽t⩽h_t,0⩽x⩽h_x,0⩽y⩽h_y,0⩽z⩽h_z\}.$$

(2)

Here $h_x$, $h_y$, $h_z$ and $h_t$ denote the spatial and temporal sizes of the volume V (Figure 1), respectively.

The mass conservation is written as

$$M\left(t\right)=M\left(0\right)+Q_{x=0}+Q_{y=0}+Q_{z=0}-Q_{x=h_x}-Q_{y=h_y}-Q_{z=h_z}+S,$$

(3)

where $M\left(0\right)$ is mass of the liquid or gas in the volume V (2) in starting time $t=0$, $M\left(t\right)$ is mass of the liquid or gas in the volume V (2) in time t$(0⩽t⩽h_t)$, $Q_{x=0}$, $Q_{y=0}$ and $Q_{z=0}$ refer to the mass of the liquid or gas going to V (2) in time t$(0⩽t⩽h_t)$ through the faces $F_{x=0}$, $F_{y=0}$ and $F_{z=0}$

$$\begin{array}{cc}\hfill F_{x=0}& =\left\{(0,y,z)\right|0⩽y⩽h_y,0⩽z⩽h_z\},\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill F_{y=0}& =\left\{(x,0,z)\right|0⩽x⩽h_x,0⩽z⩽h_z\},\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill F_{z=0}& =\left\{(x,y,0)\right|0⩽x⩽h_x,0⩽y⩽h_y\},\hfill \end{array}$$

(4c)

$Q_{x=h_x}$, $Q_{y=h_y}$ and $Q_{z=h_z}$ refer to the mass of the liquid or gas coming from V for t$(0\le t\le h_t)$ through the faces $F_{x=h_x}$, $F_{y=h_y}$ and $F_{z=h_z}$

$$\begin{array}{cc}\hfill F_{x=h_x}& =\left\{(h_x,y,z)\right|0⩽y⩽h_y,0⩽z⩽h_z\},\hfill \end{array}$$

(4d)

$$\begin{array}{cc}\hfill F_{y=h_y}& =\left\{(x,h_y,z)\right|0⩽x⩽h_x,0⩽z⩽h_z\},\hfill \end{array}$$

(4e)

$$\begin{array}{cc}\hfill F_{z=h_z}& =\left\{(x,y,h_z)\right|0⩽x⩽h_x,0⩽y⩽h_y\},\hfill \end{array}$$

(4f)

S is a source term. Let us also assume that $S=0$ and the shape of the volume V (2) remains unchanged.

The mass of the liquid or gas going to V (2) through the faces (4) for time t is

$$\begin{array}{cccccc}\hfill Q_{x=0}& =\underset{0}{\overset{t}{\int }}{m}_{x=0}\left(\xi \right)d\xi ,\hfill & \hfill Q_{y=0}& =\underset{0}{\overset{t}{\int }}{m}_{y=0}\left(\xi \right)d\xi ,\hfill & \hfill Q_{z=0}& =\underset{0}{\overset{t}{\int }}{m}_{z=0}\left(\xi \right)d\xi ,\hfill \end{array}$$

(5a)

$$\begin{array}{cccccc}\hfill Q_{x=h_x}& =\underset{0}{\overset{t}{\int }}{m}_{x=h_x}\left(\xi \right)d\xi ,\hfill & \hfill Q_{y=h_y}& =\underset{0}{\overset{t}{\int }}{m}_{y=h_y}\left(\xi \right)d\xi ,\hfill & \hfill Q_{z=h_z}& =\underset{0}{\overset{t}{\int }}{m}_{z=h_z}\left(\xi \right)d\xi ,\hfill \end{array}$$

(5b)

where $m_{x=0}\left(t\right)$, $m_{y=0}\left(t\right)$, $m_{z=0}\left(t\right)$, $m_{x=h_x}\left(t\right)$, $m_{y=h_y}\left(t\right)$ and $m_{z=h_z}\left(t\right)$ is the mass flow rates through the faces (4).

The mass balance (3) can be rewritten as

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{h_t}}\left({\displaystyle \frac{M\left(h_t\right)}{\mathrm{V}}}-{\displaystyle \frac{M\left(0\right)}{\mathrm{V}}}\right)& +{\displaystyle \frac{1}{h_t}}\underset{0}{\overset{h_t}{\int }}{\displaystyle \frac{m_{x=h_x}\left(t\right)-m_{x=0}\left(t\right)}{h_x{h}_y{h}_z}}dt\hfill \\ \hfill & +{\displaystyle \frac{1}{h_t}}\underset{0}{\overset{h_t}{\int }}{\displaystyle \frac{m_{y=h_y}\left(t\right)-m_{y=0}\left(t\right)}{h_x{h}_y{h}_z}}dt+{\displaystyle \frac{1}{h_t}}\underset{0}{\overset{h_t}{\int }}{\displaystyle \frac{m_{z=h_z}\left(t\right)-m_{z=0}\left(t\right)}{h_x{h}_y{h}_z}}dt=0,\hfill \end{array}$$

(6)

where $\mathrm{V}=h_x{h}_y{h}_z$.

The classical definition of the continuous medium density ($\rho$) is ratio of the medium mass M in the finite volume V on $\mathrm{V}=h_x{h}_y{h}_z$, if $max(h_t,h_x,h_y,h_z)\to 0$

$$\underset{V\to 0}{lim}{\displaystyle \frac{M\left(V\right)}{V}}=\rho (t,x,y,z).$$

(7)

Limiting $max(h_t,h_x,h_y,h_z)\to 0$ and taking into account that

$$\begin{array}{c}\underset{h\to 0}{lim}{\displaystyle \frac{1}{h}}\underset{0}{\overset{h}{\int }}f\left(x\right)dx=f\left(0\right),\\ \underset{h\to 0}{lim}{\displaystyle \frac{f\left(h\right)-f\left(0\right)}{h}}=f^{\prime }\left(0\right),\end{array}$$

Equation (6) becomes

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{\partial \rho }{\partial t}}\right|}_{\begin{array}{c}t=0\\ x=0\\ y=0\\ z=0\end{array}}+{\left.{\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}}\right|}_{\begin{array}{c}t=0\\ x=0\\ y=0\\ z=0\end{array}}+{\left.{\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}}\right|}_{\begin{array}{c}t=0\\ x=0\\ y=0\\ z=0\end{array}}+{\left.{\displaystyle \frac{\partial \left(\rho w\right)}{\partial z}}\right|}_{\begin{array}{c}t=0\\ x=0\\ y=0\\ z=0\end{array}}=0.\end{array}$$

Due to arbitrary choice of the finite volume V (2), this continuity equation will be valid for any point in domain where the liquid or gas flows. Therefore, the continuity equation takes the final form (1a) [15,16].

It is clear that the density cannot be defined correctly as (7). Let V and $M\left(V\right)$ be a finite volume and mass of liquid or gas in this volume, respectively. The shrinking of the finite volume to internal point results in uncertainty

$$\underset{V\to 0}{lim}{\displaystyle \frac{M\left(V\right)}{V}}=\left[{\displaystyle \frac{0}{0}}\right],$$

because $M\left(V\right)\to 0$ as $V\to 0$. In addition, the pressure and temperature of a gas are the measure of the mean momentum and the mean kinetic energy of the molecules in the finite volume, respectively. Hence, the mass conservation equation must be used in form of (6), i.e., without shrinking of the finite volume to an internal point.

The basic idea is to generate a computational grid and to formulate the governing equations over the finite volumes of this grid. The eight sets of points

$$\begin{array}{cc}\hfill {\alpha }_l^{\mathrm{v}}& ={\displaystyle \frac{l-1}{{\aleph }_{\alpha }}}=(l-1)h_{\alpha },l=1,2,\dots ,{\aleph }_{\alpha },{\aleph }_{\alpha }+1,\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill {\alpha }_l^{\mathrm{f}}& ={\displaystyle \frac{2l-1}{2{\aleph }_{\alpha }}}=(2l-1){\displaystyle \frac{h_{\alpha }}{2}},l=1,2,\dots ,{\aleph }_{\alpha }-1,{\aleph }_{\alpha },\hfill \end{array}$$

(8b)

define a Cartesian grid, where $h_{\alpha }=1/{\aleph }_{\alpha }$ and $\alpha ={\left(txyz\right)}^T$. The points $(t_n^{\mathrm{v}},x_i^{\mathrm{v}},y_j^{\mathrm{v}},z_k^{\mathrm{v}})$ are the grid vertices and the finite volume is defined by

$${\mathcal{V}}_{ijk}^{\left(n\right)}=\left\{(t,x,y,z)\right|t_{n-1}^{\mathrm{f}}<t<t_n^{\mathrm{f}},x_{i-1}^{\mathrm{f}}<x<x_i^{\mathrm{f}},y_{j-1}^{\mathrm{f}}<y<y_j^{\mathrm{f}},z_{k-1}^{\mathrm{f}}<z<z_k^{\mathrm{f}}\},$$

(9)

where the points $t_n^{\mathrm{f}}$, $x_i^{\mathrm{f}}$, $y_j^{\mathrm{f}}$ and $z_k^{\mathrm{f}}$ define the finite volume faces. The mesh size and the minimum spatial scale ${\hslash }_s$ satisfy to

$$min(h_x,h_y,h_z)\ge {\hslash }_s\gg l_{\mathrm{mol}},$$

(10)

where $l_{\mathrm{mol}}$ is the mean free path of a molecule of the liquid or gas. The minimum size of the finite volume ${\mathcal{V}}_{ijk}^{\left(n\right)}$ (9) is chosen such that

$$\mathsf{Kn}={\displaystyle \frac{l_{\mathrm{mol}}}{min(h_x,h_y,h_z)}}<{10}^{-3},$$

where $\mathsf{Kn}$ is the Knudsen number, i.e., the medium inside ${\mathcal{V}}_{ijk}^{\left(n\right)}$ is continuous.

The time step for the numerical solution of non-stationary equations should be limited by

$$h_t\ge {\hslash }_t\gg t_{mol}>0,$$

(11)

where $t_{mol}$ is the average time between collisions of two molecules and ${\hslash }_t$ is the minimum time scale.

Since all macroparameters are defined over the finite volumes (i.e., all functions are constant inside ${\mathcal{V}}_{ijk}^{\left(n\right)}$ (9)), the functions $\rho$, p, T can be assigned to any point of the volume (for example, to the vertex $(t_n^{\mathrm{v}},x_i^{\mathrm{v}},y_j^{\mathrm{v}},z_k^{\mathrm{v}})\in {\mathcal{V}}_{ijk}^{\left(n\right)}$). Due to limitations (10) and (11), the density

$${\displaystyle \frac{M\left({\mathcal{V}}_{ijk}^{\left(n\right)}\right)}{\mathrm{V}}}={\rho }_{ijk}^{\left(n\right)}$$

is defined correctly.

Note that the discontinuous conservative variables ($\rho$, p, T, etc.) are assigned to the vertices, but the continuous flux variables ($\rho u$, $\rho v$, $\rho w$, etc.) are assigned to the faces of ${\mathcal{V}}_{ijk}^{\left(n\right)}$ (9). In general, it is necessary to express the flux variables in terms of the conservative ones using some interpolation.

The linear interpolation of the flux variables and the composite method for the integral evaluation leads to final form of the law of conservation of mass over the finite volume (9)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\rho }_{ijk}^{(n+1)}-{\rho }_{ijk}^{\left(n\right)}}{h_t}}& +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\rho }_{i+1jk}^{\left(n\right)}{u}_{i+1jk}^{\left(n\right)}-{\rho }_{i-1jk}^{\left(n\right)}{u}_{i-1jk}^{\left(n\right)}}{2h_x}}+{\displaystyle \frac{{\rho }_{i+1jk}^{(n+1)}{u}_{i+1jk}^{(n+1)}-{\rho }_{i-1jk}^{(n+1)}{u}_{i-1jk}^{(n+1)}}{2h_x}}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\rho }_{ij+1k}^{\left(n\right)}{u}_{ij+1k}^{\left(n\right)}-{\rho }_{ij-1k}^{\left(n\right)}{u}_{ij-1k}^{\left(n\right)}}{2h_y}}+{\displaystyle \frac{{\rho }_{ij+1k}^{(n+1)}{u}_{ij+1k}^{(n+1)}-{\rho }_{ij-1k}^{(n+1)}{u}_{ij-1k}^{(n+1)}}{2h_y}}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\rho }_{ijk+1}^{\left(n\right)}{u}_{ijk+1}^{\left(n\right)}-{\rho }_{ijk-1}^{\left(n\right)}{u}_{ijk-1}^{\left(n\right)}}{2h_z}}+{\displaystyle \frac{{\rho }_{ijk+1}^{(n+1)}{u}_{ijk+1}^{(n+1)}-{\rho }_{ijk-1}^{(n+1)}{u}_{ijk-1}^{(n+1)}}{2h_z}}\right)=0,\hfill \end{array}$$

(12)

where $h_t=t_{n+1}^{\mathrm{v}}-t_n^{\mathrm{v}}$, $h_x=x_i^{\mathrm{f}}-x_{i-1}^{\mathrm{f}}$, $h_y=y_j^{\mathrm{f}}-y_{j-1}^{\mathrm{f}}$, $h_z=z_k^{\mathrm{f}}-z_{k-1}^{\mathrm{f}}$ are the mesh sizes, and $\mathrm{V}=h_x{h}_y{h}_z$. Since all integrable functions are piecewise constant, all integrals are evaluated exactly. Really, it is similar to the Crank–Nicolson scheme with the piecewise constant functions (the conservative variables). From a physical point of view, the explicit scheme

$${\displaystyle \frac{{\widehat{\rho }}_{ijk}^{(n+1)}-{\widehat{\rho }}_{ijk}^{\left(n\right)}}{h_t}}+{\left({\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}}\right)}_{ijk}^{\left(n\right)}+{\left({\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}}\right)}_{ijk}^{\left(n\right)}+{\left({\displaystyle \frac{\partial \left(\rho w\right)}{\partial z}}\right)}_{ijk}^{\left(n\right)}=0,$$

and the implicit scheme

$${\displaystyle \frac{{\widehat{\rho }}_{ijk}^{(n+1)}-{\widehat{\rho }}_{ijk}^{\left(n\right)}}{h_t}}+{\left({\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}}\right)}_{ijk}^{(n+1)}+{\left({\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}}\right)}_{ijk}^{(n+1)}+{\left({\displaystyle \frac{\partial \left(\rho w\right)}{\partial z}}\right)}_{ijk}^{(n+1)}=0,$$

are not accurate, since the mass transfer (${\left(\rho u\right)}_x^{\prime }+{\left(\rho v\right)}_y^{\prime }+{\left(\rho w\right)}_z^{\prime }$) occurs only at one time step. The law of conservation of mass (6) on the unstaggered grid is given in [17].

The laws of conservation of momentum and energy over the finite volume ${\mathcal{V}}_{ijk}^{\left(n\right)}$ (9) can be formulated in the same way.

3. Remarks on Phenomenological Laws

Many phenomenological laws (Fourier’s law, Fick’s law, Newton’s law and others) have a gradient-based form. Due to limitations (10) and (11), the integral form of the phenomenological laws should be used in the governing equations. For example, instead of differential Fourier’s law

$$\overrightarrow{q}=-\lambda \left(T\right)\nabla T$$

the integral form

$$q={\langle \lambda \rangle }_i{\displaystyle \frac{T_{i+1}^{\left(n\right)}-T_i^{\left(n\right)}}{x_{i+1}^{\mathrm{v}}-x_i^{\mathrm{v}}}}$$

should be used in the energy equations, where

$${\langle \lambda \rangle }_i=2{\displaystyle \frac{\lambda \left(T_i^{\left(n\right)}\right)\lambda \left(T_{i+1}^{\left(n\right)}\right)}{\lambda \left(T_i^{\left(n\right)}\right)+\lambda \left(T_{i+1}^{\left(n\right)}\right)}}.$$

is the averaging coefficient of thermal conductivity $\lambda \left(T\right)$. The interface temperature $T^{*}$ in the contact point $x^{*}=(x_i^{\mathrm{v}}+x_{i+1}^{\mathrm{v}})/2$ is defined by

$$T^{*}={\displaystyle \frac{\lambda \left(T_i^{\left(n\right)}\right)T_i^{\left(n\right)}+\lambda \left(T_{i+1}^{\left(n\right)}\right)T_{i+1}^{\left(n\right)}}{\lambda \left(T_i^{\left(n\right)}\right)+\lambda \left(T_{i+1}^{\left(n\right)}\right)}}.$$

4. Numerical Experiments

First of all, it is necessary to emphasise the difference between continuum and discontinuum approximations using the simplest 1D equations. Let an equation in the continuum mechanics

$$u^{\prime }=f\left(x\right)$$

be an exact problem and a discrete equation

$${\displaystyle \frac{u\left(x_i^{\mathrm{f}}\right)-u\left(x_{i-1}^{\mathrm{f}}\right)}{h}}=f\left(x_i^{\mathrm{v}}\right)$$

be an approximate (discrete) problem. Here, $x_i^{\mathrm{v}}$ and $x_i^{\mathrm{f}}$ are the vertices and the finite volume faces of the uniform grid (8), and h is a mesh size. The difference between the differential equation and their discrete analogue on $[x_{i-1}^{\mathrm{f}},x_i^{\mathrm{f}}]$ becomes

$$\mathcal{E}=\underset{[x_{i-1}^{\mathrm{f}},x_i^{\mathrm{f}}]}{max}\left|u^{\prime }\left(x_i^{\mathrm{v}}\right)-{\displaystyle \frac{u\left(x_i^{\mathrm{f}}\right)-u\left(x_{i-1}^{\mathrm{f}}\right)}{h}}\right|.$$

In the continuum approximation, the differential equations are assumed to be exact problems, but their discrete analogues are assumed to be approximate problems. If $u\left(x\right)$ is a sufficiently differentiable function and $|f^{\left(n\right)}\left(x\right)|<Const$, one can apply a Taylor expansion to $u\left(x_i^{\mathrm{f}}\right)-u\left(x_{i-1}^{\mathrm{f}}\right)$ with respect to h

$$\mathcal{E}\left(h\right)\approx {\displaystyle \frac{h^2}{24}}\underset{[x_{i-1}^{\mathrm{f}},x_i^{\mathrm{f}}]}{max}\left|f^{\prime }\left(x_i^{\mathrm{v}}\right)\right|.$$

The method is said to be convergent if $\mathcal{E}\left(h\right)\to 0$ as $h\to 0$, i.e., ultimately converging to the corresponding solution of the original differential equation at every vertex of the computational grid (8).

In the discontinuum approximation, the discrete equations are assumed to be exact problems, but their differential analogues as $h\to 0$ are assumed to be approximate problems. In this case, the error estimation becomes

$${\displaystyle \frac{{\hslash }_s^2}{24}}\underset{[x_{i-1}^{\mathrm{f}},x_i^{\mathrm{f}}]}{min}\left|f^{\prime }\left(x_i^{\mathrm{v}}\right)\right|⩽\mathcal{E}⩽{\displaystyle \frac{h^2}{24}}\underset{[x_{i-1}^{\mathrm{f}},x_i^{\mathrm{f}}]}{max}\left|f^{\prime }\left(x_i^{\mathrm{v}}\right)\right|,$$

where ${\hslash }_s$ is the least space scale. Since $h\ge {\hslash }_s>0$ and $f^{\left(n\right)}\left(x\right)\ne 0$, the solutions of discrete and differential problems are generally different. It means that the exact solution of the Navier–Stokes (1) is considered as an approximate solution in discontinuum mechanics.

We start our discussion with the discrete boundary value problem

$${\displaystyle \frac{u_{i-1}^h-2u_i^h+u_{i+1}^h}{h^2}}+4{\pi }^2{\omega }^{2}sin\left(2\pi \omega x_i^{\mathrm{v}}\right)=0,u_1^h=u_{\aleph +1}^h=0,$$

(13)

where $\omega =1,2,3,\dots$ is an integer parameter. The approximate boundary value problem

$$u^{″}\left(x\right)+4{\pi }^2{\omega }^{2}sin\left(2\pi \omega x\right)=0,u\left(0\right)=u\left(1\right)=0,$$

(14)

has the exact solution

$$u\left(x\right)=sin\left(2\pi \omega x\right).$$

(15)

To illustrate this convergence, the criterion

$$\chi =\underset{i=1,2,\dots ,\aleph +1}{max}\left|u\left(x_i^{\mathrm{v}}\right)-u_i^h\right|$$

(16)

is used to measure the difference between exact solutions of the approximate (14) and the discrete (13) boundary value problems. It is expected that $\chi \le C{h}^2$, where C is some h-independent constant and $h>\hslash$.

The finite volume method is the best discretisation approach for the geometric multigrid methods based on agglomeration of the finite volumes. The Robust Multigrid Technique (RMT) had been proposed and developed for parallel solving nonlinear (initial-)boundary value problems in a black-box manner [18,19]. RMT uses the multiple triple coarsening for problem-independent transfer operators. Figure 2 represents fine (l) and coarse grids ($l-1$ and $l-2$) obtained by the triple coarsening. The fine and coarse grid vertices are given by

$$x_{\left\{i\right\}}^{\mathrm{v}}=x_i^{\mathrm{v}},i=1,2,\dots ,{\aleph }_l+1,$$

where i and $\left\{i\right\}$ are the indices of the coarse and fine grid vertices

$$\left\{i\right\}=3i-2.$$

Since each function is constant inside the finite volume (Figure 2), the criterion

$$s=\underset{i=2,3,\dots ,{\aleph }_{l-2}}{max}\left|u_i^h-{\displaystyle \frac{{\overline{u}}_{\left\{i\right\}-1}^h+{\overline{u}}_{\left\{i\right\}}^h+{\overline{u}}_{\left\{i\right\}+1}^h}{3}}\right|$$

(17)

can be used to measure the grid convergence. It is easy to see that

$$s\le 10\chi .$$

Figure 3 demonstrates the function (15) and the solution of (13) as a piecewise constant function having only finitely many pieces. The grid convergence in solving the discrete boundary value problem (13) is demonstrated in Figure 4, which shows the reduction of criteria $\chi$ and s as a function of the number of grid vertices $\aleph +1$ ($h=1/\aleph$) for $\omega =2$ and $\omega =6$. The linear behaviour of S is obtained for the ℵ-range ${10}^2<\aleph <{10}^5$. It is expected that $\chi \approx S/10$ in this range, where the criterion s can be measured in computations.

A similar approach based on the mathematical formalism was proposed for Godunov-type schemes. The article shows that mathematical modelling of physical and chemical processes based on the physical formalism (physically meaningful macroparameters on the finite volumes) is more general than that based on the mathematical formalism.

The behaviour of grid convergence makes it possible to propose a robust criterion for grid refinement. Some of the grid generation issues that occur in the context of convection–diffusion equations can be illustrated by the linear 1D boundary value problem

$$\epsilon {\displaystyle \frac{u_{i-1}^h-2u_i^h+u_{i+1}^h}{h^2}}+{\displaystyle \frac{u^h\left(x_i^{\mathrm{f}}\right)-u^h\left(x_{i-1}^{\mathrm{f}}\right)}{h}}=0,u_1^h=0,u_{\aleph +1}^h=1,$$

(18)

where $0<\epsilon \ll 1$ is a small parameter. The numerical solution of this problem has a sharp gradient (boundary layer) near the boundary $x=0$, the width of the layer depends on the parameter $\epsilon$.

In a weighted-order upwind scheme, the second (convection) term is approximated by weighted linear interpolation such that only upstream and central grid points are used in the discretisation

$$u^h\left(x_i^{\mathrm{f}}\right)=(1-\gamma )u_i^h+\gamma u_{i+1}^h,$$

where $\gamma \in [0,1]$ is the weighted parameter. Equation (18) can be rewritten in matrix form

$$A{u}_{i-1}^h+B{u}_i^h+C{u}_{i+1}^h=0,i=2,3,\dots ,\aleph ,$$

where

$$A={\displaystyle \frac{\epsilon }{h}}+\gamma -1,B=-2{\displaystyle \frac{\epsilon }{h}}-2\gamma +1,C={\displaystyle \frac{\epsilon }{h}}+\gamma ,\gamma =max\left({\displaystyle \frac{1}{2}},1-{\displaystyle \frac{\epsilon }{h}}\right).$$

The approximate boundary value problem

$$\epsilon u^{″}+u^{\prime }=0,u\left(0\right)=0,u\left(1\right)=1,$$

has the exact solution

$$u\left(x\right)={\displaystyle \frac{1-exp(-\frac{x}{\epsilon })}{1-exp(-\frac{1}{\epsilon })}}.$$

(19)

The exact solution (19) of the differential problem is defined in the domain $[0,1]$. The classical numerical solution is defined at the grid vertices $x_i^{\mathrm{v}}$. The piecewise constant solution is defined over the domain $[0,1]$. Therefore, it is possible to compare the numerical solutions obtained on different grids over the domain $[0,1]$. Figure 5 represents the convergence history for different mesh sizes $h=1/\aleph$. The small value of the parameter s shows that the numerical solution of (18) on the coarse grids ($\aleph =10$ and $\aleph =30$) does not have the boundary layer near $x=0$. The refinement of grid ($\aleph =90$) shows an increase in the parameter s value. This means that the numerical solution of (18) has the boundary layer near $x=0$, but the accuracy of computations inside the layer is low. The linear decrease of s ($\aleph >270$) shows that the accuracy of numerical solution depends only on the mesh size $h=1/\aleph$ (Figure 5). Figure 6 represents all solutions of the model convection–diffusion problem.

The criterion s (17) makes it possible to generate the adaptive grids in a black-box manner [20,21,22,23]. In addition, a robust stopping criterion based on the parameter s value can be proposed for multigrid methods [24].

5. From Boltzmann Equation to Equations of the Computational Macromechanics

The derivative-free description can be used for other phenomena [25,26]. Consider the kinetic Boltzmann equation for the expansion of the derivative-free approach. Let us define the distribution function $f(\overrightarrow{\mathit{x}},\overrightarrow{\mathit{v}},t)$ such that $f(\overrightarrow{\mathit{x}},\overrightarrow{\mathit{v}},t)d^{3}x{d}^{3}v$ is probability of finding a particle in phase space volume $d^{3}x{d}^{3}v$ centred on $\overrightarrow{\mathit{x}}$, $\overrightarrow{\mathit{v}}$ at time t. The Boltzmann equation becomes

$$f_t^{\prime }+{\overrightarrow{\mathbf{\nabla }}}_{\mathit{x}}·\left(\overrightarrow{\mathit{v}}f\right)+{\overrightarrow{\mathbf{\nabla }}}_{\mathit{v}}·\left(\overrightarrow{\mathit{g}}f\right)=J,$$

where J represents discontinuous motion of particles through phase space because of collisions. Let X be a finite volume in physical space. The particle velocity space can be discretised in the same way. The volume-averaged density is defined by

$$\rho ({\mathit{x}}_{*},t)=\underset{V}{\int }{\displaystyle \frac{1}{X}}\underset{X}{\int }mf(\overrightarrow{\mathit{x}},\overrightarrow{\mathit{v}},t)d^{3}x{d}^{3}v,$$

where ${\mathit{x}}_{*}\in X$ and m is the particle mass. In the simplest case of the same particle, local mass conservation leads to $J=0$ and the Boltzmann equation multiplied by m becomes

$${\displaystyle \frac{\partial \left(mf\right)}{\partial t}}+\sum _{i=1}^3{\displaystyle \frac{\partial \left(mf{v}_i\right)}{\partial x_i}}+\sum _{i=1}^3{\displaystyle \frac{\partial \left(mf{g}_i\right)}{\partial x_i}}=0.$$

Integration over the finite volume in the phase space and over $t^{(n+1)}-t^{\left(n\right)}$ in time leads to the following volume-averaged form of the Boltzmann equation

$$\underset{V}{\int }{\displaystyle \frac{1}{X}}\underset{X}{\int }{\displaystyle \frac{1}{t^{(n+1)}-t^{\left(n\right)}}}\underset{t^{\left(n\right)}}{\overset{t^{(n+1)}}{\int }}{\displaystyle \frac{\partial \left(mf\right)}{\partial t}}dt{d}^{3}x{d}^{3}v$$

$$+\sum _{i=1}^3\underset{V}{\int }{\displaystyle \frac{1}{X}}\underset{X}{\int }{\displaystyle \frac{1}{t^{(n+1)}-t^{\left(n\right)}}}\underset{t^{\left(n\right)}}{\overset{t^{(n+1)}}{\int }}\left({\displaystyle \frac{\partial \left(mf{v}_i\right)}{\partial x_i}}+{\displaystyle \frac{\partial \left(mf{g}_i\right)}{\partial x_i}}\right)dt{d}^{3}x{d}^{3}v=0.$$

Using the divergence theorem, the second term can be written as

$$\underset{V}{\int }{\displaystyle \frac{1}{X}}\underset{X}{\int }{\displaystyle \frac{m}{2}}\left({\overrightarrow{\mathbf{\nabla }}}_{\mathit{x}}·{\left(\overrightarrow{\mathit{v}}f\right)}^{\left(n\right)}+{\overrightarrow{\mathbf{\nabla }}}_{\mathit{x}}·{\left(\overrightarrow{\mathit{v}}f\right)}^{(n+1)}\right)d^{3}x{d}^{3}v$$

$$+{\displaystyle \frac{1}{2X}}\underset{X}{\int }\underset{V}{\int }m\left({\overrightarrow{\mathbf{\nabla }}}_{\mathit{v}}·{\left(\overrightarrow{\mathit{g}}f\right)}^{\left(n\right)}+{\overrightarrow{\mathbf{\nabla }}}_{\mathit{v}}·{\left(\overrightarrow{\mathit{g}}f\right)}^{(n+1)}\right)d^{3}v{d}^{3}x$$

$$=\underset{V}{\int }{\displaystyle \frac{1}{2X}}\underset{S_x}{\int }{\overrightarrow{\mathit{n}}}_x·\left({\left(m\overrightarrow{\mathit{v}}f\right)}^{\left(n\right)}+{\left(m\overrightarrow{\mathit{v}}f\right)}^{(n+1)}\right)dA+{\displaystyle \frac{1}{2X}}\underset{X}{\int }\underset{S_v}{\int }{\overrightarrow{\mathit{n}}}_v·\left({\left(m\overrightarrow{\mathit{g}}f\right)}^{\left(n\right)}+{\left(m\overrightarrow{\mathit{g}}f\right)}^{(n+1)}\right)dB.$$

Finally, the volume-averaged conservation equation of mass

$${\displaystyle \frac{\rho ({\mathit{x}}_{*},t^{(n+1)})-\rho ({\mathit{x}}_{*},t^{\left(n\right)})}{t^{(n+1)}-t^{\left(n\right)}}}$$

$$+{\displaystyle \frac{1}{2X}}\underset{S_x}{\int }\left(\rho \left({\mathit{x}}_{*},t^{\left(n\right)}\right){\overrightarrow{\mathit{V}}}_n\left({\mathit{x}}_{*},t^{\left(n\right)}\right)+\rho \left({\mathit{x}}_{*},t^{(n+1)}\right){\overrightarrow{\mathit{V}}}_n\left({\mathit{x}}_{*},t^{(n+1)}\right)\right)dA=0$$

can be obtained from the Boltzmann equation averaged over the finite volumes in physical space.

A more attractive approach is direct modelling in a discretised space [27]. From the point of view of physical modelling, the derivative-free Boltzmann equation is more general than the classical one because the continuity assumption is not needed for flow variables.

6. Discussion

The classical mathematical description of physical and chemical processes in the continuum approximation is based on the mutually contradictory use of differential calculus and minimal spatial and temporal scales applied to ignore the molecular structure of matter. The resulting governing (partial) differential equations (PDEs) connect physically senseless functions assigned to points $(t,x,y,z)$. The differentiability or continuity assumption is needed for all variables in the governing PDEs. The numerical solution of these PDEs implies the transformation of (initial-)boundary value problems into algebraic ones, i.e., to approximate them by algebraic equations that involve a finite number of unknowns. There are two the most popular ways to discretise PDEs. The finite element method is based on replacing the original function with a function that has some degree of smoothness over the global domain, but is piecewise polynomial on simple subdomains. The finite volume method is based on the approximation of continuous conservation laws [10].

Now it is necessary to completely abandon the use of differential calculus for the mathematical modelling of physical and chemical processes in the continuum approximation [28]. For these purposes, the global domain is divided into a number of finite volumes of such size that the liquid or gas in them can be considered continuous. In this case, all variables (density, pressure, temperature, etc.) have a physical sense in strict accordance with theoretical definitions, and therefore they are not assigned to points, but to finite volumes. As a result, all physical and chemical processes can be described only in a finite volume-averaging sense, i.e., all variables are piecewise constant functions (so-called discontinuum approximation: constant functions in the finite volume and discontinuous on its faces). The mathematical models of physical and chemical processes in each finite volume consist of fundamental conservation laws, discrete phenomenological laws and additional hypotheses. Really, the finite volume schemes are the mathematical models in the discontinuum approximation. The derivative-free description of physical and chemical processes is very preferable from a computing point of view. All variables must be integrable functions, they are not required to be differentiable. This is especially noticeable for the Godunov-type schemes [29,30]. From a physical point of view, the explicit and implicit schemes cannot be used for mathematical modelling.

The current target of computational mathematics is to recover the solution of original PDEs as the mesh size and time step approach zero. This leads to physically meaningless macroparameters (density, pressure, temperature and others defined over a finite volume of a continuous medium), since there are minimum spatial and temporal scales for each model. The minimum mesh size and time step cannot be less than the minimum spatial and temporal scales. This leads to derivative-free models. Then differential equations form an approximate model, the discrete equations form an exact model, and the finite difference, the finite volume and the finite element methods become unnecessary. A similar approach based on the mathematical formalism was proposed for Godunov-type schemes [29,30]. The article shows that mathematical modelling of physical and chemical processes based on the physical formalism (physically meaningful macroparameters on finite volumes) is more general than modelling based on the mathematical formalism.

The governing equations such as (12) in the above-mentioned discontinuous approximation depend on the shape of finite volume. To clarify, the formal form of the continuity equation

$${\displaystyle \frac{1}{t^{(n+1)}-t^{\left(n\right)}}}\underset{t^{\left(n\right)}}{\overset{t^{(n+1)}}{\int }}{\displaystyle \frac{1}{{\mathrm{V}}_{ijk}^{(n+1)}}}\underset{{\mathcal{V}}_{ijk}^{(n+1)}}{\int }\left({\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \overrightarrow{\mathit{V}}\right)\right)dvdt=0$$

can be used in theoretical analysis, where integration is performed under the condition that all functions are constant in each finite volume ${\mathcal{V}}_{ijk}^{(n+1)}$.

7. Conclusions

The dilemma of correctly defined derivatives versus correctly defined macroparameters presents itself. In the correct derivatives approach, one tries to use differential and integral calculus in order to analyse the differential problems related to physically senseless variables. On the other hand, the idea of a physically meaningful solver based on the correctly defined macroparameters is to use conservation and phenomenological laws for a derivative-free description of the physical and chemical processes on each finite volume. The mesh size and time step are limited by the continuity hypothesis. The numerical solution of the governing discrete equations is the piecewise constant functions (density, pressure, temperature, etc.) corresponding to the classic physical theories. From a practical point of view, this approach results in impressive simplification of the mathematical modelling due to the absence of discretisation of the (initial-)boundary value problems. The best solver of these governing finite volume-based equations is the agglomeration multigrid.

Future work will be an analysis of the discretisation error of the governing equations in the discontinuous approximation, solution of 2D benchmark problems of CFD and comparison with the continuum approximation.