Some Observations on Thermodynamic Basis of Pressure Continuum Condition and Consequences of Its Violation in Discretised CFD

Abstract

CFD is concerned with solution of Navier–Stokes (NS) equations in discretised space. It is important, therefore, to ensure that the discretised equations and their solutions obey the continuum condition embedded in Stokes’s stress–strain laws for an isotropic continuum fluid. In this paper, it is shown that adherence to this condition leads to three important conceptual/algorithmic outcomes: 1. Prevention of zig-zag pressure distribution when NS equations are solved for incompressible flow of a single fluid on colocated grids. 2. Prevention of loss of volume/mass at large times when NS equations are solved for interfacial incompressible flows of multi-fluids within single-fluid formalism. 3. Evaluation of surface tension force in interfacial flows without using phenomenology embedded in the definition of the surface tension coefficient. All the above benefits are justified on the basis of a thermodynamic principle rarely invoked in discretised CFD. A few problems are solved by way of case studies.

Nomenclature  

$A_{k},AP$:

Coefficients in Discretised Equations

D:

Mass Diffusivity

F:

Volume Fraction

Fr:

Froude Number

$F_{st}$:

Surface Tension Force

g:

Gravity Acceleration

k:

Thermal Conductivity

Kn:

Knudsen Number

Pc:

Peclet Number

p:

Pressure

q:

Continuum-Preserving Stress/Pressure

R:

Residual or Gas Constant

Re:

Reynolds Number

Ra:

Rayleigh Number

t:

Time

$u_{i}$:

Velocity in $x_{i}$, i = 1, 2, 3 direction

V:

Volume

$\vec {V}$:

Total Velocity Vector

We:

Weber Number

 

Greek Symbols  

$\alpha , \beta$:

Under-Relaxation Factors

$\gamma$:

Second viscosity coefficient

$\mu$:

Dynamic viscosity

$\rho$:

Density

$\sigma$:

Surface Tension Coefficient or Stress

$\tau$:

Stress

 

Suffixes  

a:

Refers to Heavier Fluid

b:

Refers to Lighter Fluid or to Boundary Node

$\text {cont}$:

Refers to continuum

$\text {disc}$:

Refers to discretised space

f:

Refers to CV Face Location

i:

In i-direction

m:

Refers to mass conservation or to mixture

n:

Normal to the Interface

sm:

Refers to Smoothing

th:

Thermodynamic

$x_{i}$:

Refers to $x_{i}, i = 1, 2, 3$ directions

 

Superscripts  

l:

Iteration Number

o:

Refers to old time

$u_{i},u_{f,i}$:

Refers to Momentum Equation

−:

Refers to Multidimensional Average

$'$:

Refers to Correction

 

Acronyms and Short Forms  

CV:

Control Volume

CFD:

Computational Fluid Dynamics

$F( \,\,\,)$:

Function of

LS:

Level-Set Method

VOF:

Volume of Fluid Method

 

1 Introduction

1.1 Navier–Stokes Equations

In the study of transport in moving fluids, the fundamental laws of motion (conservation of mass and Newton’s second law of motion) are applied to an elemental fluid. Two approaches are possible: (a) Particle approach and (b) Continuum approach.

In the particle approach, the fluid is assumed to consist of particles (molecules, atoms) and the laws are applied to study particle motion. Fluid motion is then described by statistically averaged motion of a group of particles. For most applications arising in engineering and the environment, however, this approach is too cumbersome¹ because the significant dimensions of the flow are considerably bigger than the mean free path length between molecules. In the continuum approach, therefore, statistical averaging is assumed to have been already performed and the fundamental laws are applied to portions of fluid (or control volumes) that contain a large number of particles. The information lost in averaging must, however, be recovered. This is done by invoking some further auxiliary laws and by empirical specifications of transport properties such as viscosity ($\mu$), thermal conductivity (k) and mass diffusivity (D). The transport properties are typically determined from experiments. Notionally, the continuum approach is very attractive because one can now speak of temperature, pressure or velocity at a point and relate them to what is measured by most practical instruments.

Guidance for deciding whether particle or continuum approach is to be used can be had from Knudsen number Kn = l /L, where l is the mean free path length between molecules and L is a characteristic dimension (say, the radius of a pipe) of the flow. When Kn is very small (${<} 10^{-4}$), continuum approach is considered valid. In macro-engineering and environmental flows, therefore, continuum approach is adopted.

Control Volume (CV): The CV may be defined as a region in space across the boundaries of which matter, energy and momentum may flow, and it is a region within which source or sink of the same quantities may prevail. Further, it is a region on which external forces may act.

The Navier–Stokes equations are derived by applying the law of conservation of mass and Newton’s second law of motion to a CV shown in Fig. 1. The CV having dimensions $\Delta x_{1}$, $\Delta x_{2}$ and $\Delta x_{3}$ is located at ($x_{1}, x_{2}, x_{3}$) from a fixed origin. The statements of the laws yield algebraic equations of mass and momentum conservation. These statements are then converted to partial differential equations by letting $\Delta x_{1}$, $\Delta x_{2}$ and $\Delta x_{3} \rightarrow 0$ followed by invoking the mathematical definition of a derivative in a continuum. Thus,

$$\begin{aligned} \frac{\partial ( \rho _{m} )}{\partial t} + \frac{\partial ( \rho _{m}\,u_{fj} )}{\partial x_{j}}= & {} 0 \end{aligned}$$

(1)

$$\begin{aligned} \frac{\partial ( \rho _{m}\,u_{i} )}{\partial t} + \frac{\partial }{\partial x_{j}} ( \rho _{m}\,u_{fj}\,u_{i} )= & {} \frac{\partial \sigma _{xi}}{\partial x_{i}} + \frac{\partial }{\partial x_{j}}\,\left\{ \tau _{ij}\,( 1 - \delta _{ij} ) \right\} + \rho _{m}\,B_{i} \end{aligned}$$

(2)

$$\begin{aligned} \tau _{ij}= & {} \mu \,\left[ \frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} \right] \end{aligned}$$

(3)

Fig. 1

Eulerian specifications of the control volume (CV)

where $\rho _{m}$ is fluid (or mixture) density, $u_{fj}$ are CV face velocities, $u_{j}$ are representative CV velocities and $B_{i}$ are volumetric body forces such as Buoyancy or Centrifugal or Coriolis force. Equation 3 expresses Stokes’s law connecting surface stress $\tau _{ij}$ to the co-planar strain rate via fluid property $\mu$. Finally, $\sigma _{xi}$ are total surface-normal (tensile) stresses and are modelled as [27]

$$\begin{aligned} \sigma _{x_{i}}\equiv & {} -\,p + \sigma ^{'}_{xi} = -\,( p - q ) + \tau _{ii} = -\,( p - q ) + 2\,\mu \,\frac{\partial u_{i}}{\partial x_{i}} \nonumber \\= & {} -\,( p - q ) + 2\,\mu \,\bigtriangledown \,.\,\vec {V_{f}} \end{aligned}$$

(4)

where pressure p is compressive, normal stress $\tau _{ii}$ are tensile and $\sigma ^{'}$ is called the deviatoric stress. The significance of the newly introduced quantity q in its definition requires further elaboration.

1.2 Stokes’s Continuum Condition

In Date [6], a quantity $\overline{p}$ is defined² as

$$\begin{aligned} \overline{p} \equiv -\,\frac{1}{3}\,(\sigma _{x_{1}} + \sigma _{x_{2}} + \sigma _{x_{3}}) \end{aligned}$$

(5)

Now, an often overlooked requirement of the Stokes’s relations (with or without variable properties and in the absence of relaxation processes at the molecular level [27]) is that $\overline{p}$ must equal the point value of pressure p and the latter, in turn, must equal thermodynamic pressure $p_{\text {th}}$. Thus, using Eq. 4, it follows that

$$\begin{aligned} \overline{p} = p = p_{\text {th}} = ( p - q ) - \frac{2}{3}\,\mu \,\bigtriangledown \,.\,\vec {V_{f}} \end{aligned}$$

(6)

Now, to obey the above equality, q must be appropriately chosen in continuum as well as in discretised space. We now consider the following three cases:

• 1. Case 1: ($\vec {V}$ = 0) In this hydrostatic case,

  $$\begin{aligned} \overline{p} = p - q \end{aligned}$$

  (7)

  But in this case, p can only vary linearly with $x_{1}, x_{2}, x_{3}$ and, therefore, the point value of p exactly equals its space averaged value $\overline{p}$ in both continuum and discretised space and hence

  $$\begin{aligned} q = q_{\text {cont}} = q_{\text {disc}} = 0 \rightarrow \text{ exactly } \end{aligned}$$

  (8)

• 2. Case 2: ($\mu$ = 0 or $\bigtriangledown \,.\,V_{f}$ = 0)

  Clearly when $\mu$ = 0 (inviscid flow) or $\bigtriangledown \,.\,V_{f} = {\partial u_{fi}} / {\partial x_{i}} = 0$ (constant density incompressible flow) $p = \overline{p}$ (Eq. 6) in a continuum, and hence $q_{\text {cont}} = 0$ exactly.³ But, in this case, since fluid motion is considered, p can vary arbitrarily with $x_{1}, x_{2}, x_{3}$ and, therefore, p may not equal $\overline{p}$ in a discrete space. To understand this matter, consider a case in which pressure varies arbitrarily in $x_{1}$ direction whereas its variation in $x_{2}$ and $x_{3}$ directions is constant or linear (as in a hydrostatic case). Such a variation is shown in Fig. 2. Now consider a point P. According to Stokes’s requirement, $p_{P}$ must equal $\overline{p}_{P}$ in a continuum. But, in a discretised space, the values of pressure are available at points E and W only, and if these points are equi-distant from P then $\overline{p}_{P} = 0.5\,( p_{W} + p_{E} )$. Now, this $\overline{p}_{P}$ will not equal $p_{P}$ as seen from the figure and, therefore, the requirement of the Stokes’s relations is not met. However, without violating the continuum requirement, we may set

  $$\begin{aligned} q = q_{\text {cont}} = q_{\text {disc}} = \lambda \,( p - \overline{p} ) \end{aligned}$$

  (9)

  where $\lambda$ is an arbitrary constant. In most textbooks, where continuum is assumed, $\lambda$ is trivially set to zero because p = $\overline{p}$ (Eq. 6) in a continuum.

• 3. Case 3: ($\mu \ne$ 0 or $\bigtriangledown \,.\,V_{f} \ne$ 0)

  This case represents either compressible flow where density is a state function of both temperature and pressure or an incompressible flow with density dependent on temperature or any other scalar (e.g. void or volume fraction). Thus, in this case, Stokes’s requirement will be satisfied (see Eq. 6) if we set

  $$\begin{aligned} q = \lambda \,( p - \overline{p} ) + \gamma \,\bigtriangledown \,.\,V_{f} \rightarrow \gamma = -\frac{2}{3}\,\mu \end{aligned}$$

  (10)

  where $\gamma$ is the well-known second viscosity coefficient whose value is routinely set to $-(2/3)\mu$ even in a continuum. It is instructive to note the reason for this setting. Because, if this was not done then by combining Eqs. 10 and 6, it can be shown that

  $$\begin{aligned} ( 1 - \lambda )\,( p - \overline{p} )\,\bigtriangledown \,.\,V_{f} = \left( \gamma + \frac{2}{3}\,\mu \, \right) \,( \bigtriangledown \,.\,V_{f} )^{2} \end{aligned}$$

  (11)

  Clearly, this equation suggests that the system will experience dissipation (or reversible work done at finite rate since $\bigtriangledown \,.\,V_{f}$ is associated with the rate of volume change) even in an isothermal flow [27, 36]. This is, of course, highly improbable.⁴

Fig. 2

1D variation of pressure and Stokes’s requirement

1.3 Thermodynamic Explanations

This improbability has been explained by Schlichting [27] by considering the case of an isolated spherical mass of isothermal fluid subjected to uniform normal stress $\overline{\sigma } = -\overline{p}$ (see Fig. 3a). Now, if $\gamma$ is not set to $-(2/3)\mu$, the sphere will undergo radial oscillations of compression and expansion. Here, however, we consider an alternative arrangement that will yield one further interpretation.

Fig. 3

Consequences of violating Stokes’s condition a isolated spherical mass of fluid [27], b insulated piston cylinder

Figure 3b shows the piston-cylinder arrangement typically used in undergraduate thermodynamics. The system fluid is held in a leakproof adiabatic cylinder at temperature T and pressure p. The system is in equilibrium. Now, suppose the unlikely circumstance in which external pressure $\overline{p}$ exceeds system pressure p. It is obvious that the piston will move downwards compressing the fluid. But, if we now require that the temperature of the fluid must remain constant at T (this is analogous the Schlichting’s isothermal flow), then clearly, from first law of thermodynamics, no change in internal energy $\Delta U$ is permitted in an adiabatic cylinder, and hence there can only be two consequences.

• 1. The piston will instantly bounce back to its original position to restore equilibrium. The process may repeat resulting in oscillations of the piston in time. However, in a steady-state problem, these oscillations will manifest as spatially zig-zag pressure.

• 2. Alternatively, to maintain constant internal energy, some fluid must somehow leak out although the piston is leakproof.

It is obvious that both these occurrences are improbable but are nonetheless encountered in discretised CFD when Stokes’s continuum condition (6) is violated as will be shown in Sects. 2 and 3. Incidentally, the second circumstance above is a new interpretation associated with the violation of the Stokes’s condition.

Thus, as shown in Eq. 10, the Stokes’s relations for normal stress require modifications even in a continuum when compressible flow is considered (requiring introduction of the so-called second viscosity coefficient) and a physical explanation for this modification is found from Thermodynamics. Now, the same interpretation can be afforded to $\lambda \,( p - \overline{p} )$ part of q in Eqs. 9 or 10. This term represents a necessary modification in a discretised space. This is an important departure from the forms of normal stress expressions given in standard textbooks on fluid mechanics.

1.4 Appropriate Forms of N-S Equations

Finally, from the above discussion, it is clear that the Navier–Stokes equations written out for finite-volume discretisation should preferably read as

$$\begin{aligned} \frac{\partial ( \rho _{m} )}{\partial t} + \frac{\partial ( \rho _{m}\,u_{fj} )}{\partial x_{j}}= & {} 0 \end{aligned}$$

(12)

$$\begin{aligned} \frac{\partial ( \rho _{m}\,u_{i} )}{\partial t} + \frac{\partial }{\partial x_{j}} ( \rho _{m}\,u_{fj}\,u_{i} )= & {} -\,\frac{\partial ( p - q )}{\partial x_{i}} + \frac{\partial }{\partial x_{j}}\,\left\{ \tau _{ij} \right\} + \rho _{m}\,B_{i} \end{aligned}$$

(13)

where, in the most general case, q stands for

$$\begin{aligned} q = \lambda \,( p - \overline{p} ) - \frac{2}{3}\,\mu \times \bigtriangledown \,.\,\vec {V_{f}} \end{aligned}$$

(14)

Of course, $\lambda = 0$ in a continuum but finite⁵ in a discretised space. Note, however, that in the latter, as mesh size is reduced, $p \rightarrow \overline{p}$, and hence $q_{\text {disc}} \rightarrow q_{\text {cont}}$. Also, it is important to note that since $\overline{p}$ must equal point value of pressure p in a continuum (see Eq. 6), the former must essentially correspond to the hydrostatic or spatially linear variation of pressure (in a local sense) irrespective of the flow considered. Mathematically, therefore, we may define $\overline{p}$ as

$$\begin{aligned} \overline{p} = - \frac{1}{3}\,\sum _{i=1}^{3}\,\sigma _{x_{i}} = \frac{1}{3}\,\sum _{i=1}^{3}\,\overline{p}_{x_{i}} \end{aligned}$$

(15)

where $\overline{p}_{x_{i}}$ are each a solution⁶ to

$$\begin{aligned} \frac{\partial ^{2} p}{\partial x^{2}_{i}} = 0 \end{aligned}$$

(16)

This manner of evaluation of $\overline{p}$ can be implemented on both structured and unstructured meshes [4,5,6,7, 22, 23] in discretised CFD.

2 Computations on Colocated Grids

2.1 Pressure-Correction Equation on Colocated Grids

Since it has been known that in the SIMPLE algorithm [19], zig-zag pressure prediction is avoided by the use of staggered grid arrangement of pressure and the velocity variables (see Fig. 4a), it is obvious that the pressure-correction equation applicable to colocated grids must mimic the main features of the staggered grid practice. Thus, we begin by stating that in a fully implicit iterative procedure, the cell-face velocity will be calculated from

$$\begin{aligned} u^{l+1}_{fi} = \frac{\alpha }{AP^{u_{fi}}}\,\left[ \sum _{k}A_{k}u^{l+1}_{fi,k} - \Delta V\,\frac{\partial p^{l+1}}{\partial x_{i}} \right] + (1 - \alpha )\,u^{l}_{fi} \end{aligned}$$

(17)

where $AP^{u_{fi}} = \sum \,A_{k} + \rho ^{0}_{m}\,\Delta V/\Delta t$. Now, this velocity field must satisfy mass conservation equation (1). Thus

$$\begin{aligned} \frac{\partial ( \rho ^{l+1}_{m} )}{\partial t} + \frac{\partial ( \rho ^{l+1}_{m}\,u^{l+1}_{fi} )}{\partial x_{i}} = 0 \end{aligned}$$

(18)

After substituting Eq. 17 in Eq. 18, we make use of following representations:

$$\begin{aligned} u^{l+1}_{fi} = u^{l}_{fi} + u^{'}_{fi} \text{ and } p^{l+1} = p^{l} + p^{'}_{m} \end{aligned}$$

(19)

The above operations result⁷ in

$$\begin{aligned} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l+1}_{m}\,D_{i}\, \frac{\partial p^{'}_{m}}{\partial x_{i}} \right]= & {} \frac{\partial ( \rho ^{l+1}_{m} )}{\partial t} + \frac{\partial ( \rho ^{l+1}_{m}\,u^{l}_{fi} )}{\partial x_{i}} \nonumber \\- & {} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l+1}_{m}\,D_{i}\,R_{ufi} \right] \rightarrow D_{i} = \frac{\alpha \, \Delta V}{Ap^{u_{fi}}} \end{aligned}$$

(20)

where $R_{ufi}$ is the residual per unit volume and is given by

$$\begin{aligned} R_{ufi} = \frac{AP^{ufi}\,u^{l}_{fi} - \sum \,A_{k}u^{l}_{fi,k}}{\Delta V} + \frac{\partial p^{l}}{\partial x_{i}} \end{aligned}$$

(21)

Fig. 4

a Staggered grid arrangement of variables and associated CVs, b colocated grids—cell faces are midway between the adjacent nodes

2.2 Analysis of Discretised Equation

To understand further developments of this paper, following comments are now pertinent:

• 1. On both staggered and colocated grids, the pressure is stored at node P and the mass conservation equation is solved over the control volume surrounding node P. Therefore, with reference to Fig. 4b, the discretised version of Eq. 20 in two dimensions will read as

  $$\begin{aligned} AP\,p^{'}_{m, P}= & {} AE\,p^{'}_{m, E} + AW\,p^{'}_{m, W} + AN\,p^{'}_{m, N} + AS\,p^{'}_{m, S} \nonumber \\- & {} \dot{m}_{P} + \dot{m}_{R} \end{aligned}$$

  (22)

  where

  $$\begin{aligned} AE= & {} \frac{\rho ^{l+1}_{m}\,\alpha \,\Delta x^{2}_{2}}{AP^{uf1}}\,|_{e}, AW = \frac{\rho ^{l+1}_{m}\,\alpha \,\Delta x^{2}_{2}}{AP^{uf1}}\,|_{w} \nonumber \\ AN= & {} \frac{\rho ^{l+1}_{m}\,\alpha \,\Delta x^{2}_{1}}{AP^{uf2}}\,|_{n}, AS = \frac{\rho ^{l+1}_{m}\,\alpha \,\Delta x^{2}_{1}}{AP^{uf2}}\,|_{s} \nonumber \\ AP= & {} AE + AW + AN + AS \end{aligned}$$

  (23)

  Further, the mass source $\dot{m}_{P}$ and residual source $\dot{m}_{R}$ will be given by

  $$\begin{aligned} \dot{m}_{P}= & {} ( \rho ^{l+1}_{m}\,u^{l}_{f1}\,|_{e} - \rho ^{l+1}_{m}\,u^{l}_{f1}\,|_{w} )\,\Delta x_{2} \nonumber \\+ & {} ( \rho ^{l+1}_{m}\,u^{l}_{f2}\,|_{n} - \rho ^{l+1}_{m}\,u^{l}_{f2}\,|_{s} )\,\Delta x_{1} \nonumber \\+ & {} ( \rho ^{l+1}_{m,P} - \rho ^{o}_{m,P} )\,\frac{\Delta V}{\Delta t} \end{aligned}$$

  (24)
  $$\begin{aligned} \dot{m}_{R}= & {} AE\,R_{uf1}\,\Delta x_{1}|_{e} - AW\,R_{uf1}\,\Delta x_{1}|_{w} \nonumber \\+ & {} AN\,R_{uf2}\,\Delta x_{2}|_{n} - AS\,R_{uf2}\,\Delta x_{2}|_{s} \end{aligned}$$

  (25)

• 2. On staggered grids, momentum equations are solved at the cell faces and, therefore, residuals $R_{uf1}$ and $R_{uf2}$ must vanish at full convergence rendering $\dot{m}_{R} = 0$. Although this state of affairs will prevail only at convergence, one may ignore $\dot{m}_{R}$ even during iterative solution. Thus, effectively, Eq. 20 applicable to staggered grid arrangement is

  $$\begin{aligned} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l+1}_{m}\,D_{i}\, \frac{\partial p^{'}_{m}}{\partial x_{i}} \right]= & {} \frac{\partial ( \rho ^{l+1}_{m} )}{\partial t} + \frac{\partial ( \rho ^{l+1}_{m}\,u^{l}_{fi} )}{\partial x_{i}} \end{aligned}$$

  (26)

  The above equation is solved with the boundary condition [19]

  $$\begin{aligned} \frac{\partial p^{'}_{m}}{\partial n}\,|_{b} = 0 \end{aligned}$$

  (27)

  If the boundary pressure $p_{b}$ is specified then, of course, $p^{'}_{m,b} = 0$.

• 3. When computing on colocated grid, however, cell-face velocities must be evaluated by interpolation to complete evaluation of $\dot{m}_{P}$ because only nodal velocities $u_{i, P}$ are computed through momentum equations. Thus, $\dot{m}_{P}$ in Eq. 24 is evaluated as

  $$\begin{aligned} \overline{\dot{m}}_{P}= & {} ( \rho ^{l+1}_{m}\,\overline{u}^{l}_{1}\,|_{e} - \rho ^{l+1}_{m}\,\overline{u}^{l}_{1}\,|_{w} )\,\Delta x_{2} \nonumber \\+ & {} (\rho ^{l+1}_{m}\,\overline{u}^{l}_{2}\,|_{n} - \rho ^{l+1}_{m}\,\overline{u}^{l}_{2}\,|_{s} )\,\Delta x_{1} \nonumber \\+ & {} ( \,\rho ^{l+1}_{m,P} - \rho ^{o}_{m,P} \,)\,\frac{\Delta V}{\Delta t} \end{aligned}$$

  (28)

  where the mean velocities $\overline{u}_{i}$ are evaluated by one-dimensional averaging although multidimensional averaging can also be preferred. Thus, since the cell faces are midway between the nodes, we may write

  $$\begin{aligned} \overline{u}_{1,e}= & {} \frac{1}{2}\,(u_{1,P} + u_{1,E}) \overline{u}_{1,w} = \frac{1}{2}\,(u_{1,P} + u_{1,W}) \nonumber \\ \overline{u}_{2,n}= & {} \frac{1}{2}\,(u_{2,P} + u_{2,N}) \overline{u}_{2,s} = \frac{1}{2}\,(u_{2,P} + u_{2,S}) \end{aligned}$$

  (29)

  Replacing $u_{f1,e}$ by $\overline{u}_{1,e}$, etc. in the above manner, of course, does not guarantee that $\dot{m}_{P}$ will vanish even at convergence.⁸

• 4. Similarly, to evaluate $\dot{m}_{R}$ from Eq. 25, we reconsider Eq. 21 for cell-face location e, for example, and write it as

  $$\begin{aligned} R_{u_{f1, e}} = \overline{\frac{AP^{u_{f1}}\,u^{l}_{f1} - \sum \,A_{k}\,u^{l}_{f1,k} }{\Delta V}}\,|_{e} + \frac{\partial p^{l}}{\partial x_{1}}\,|_{e} \end{aligned}$$

  (30)

  In this equation, the net-momentum-transfer terms are now multidimensionally averaged. This is necessary because when computing on colocated grids, coefficients $A_{k}$ are not available at the cell-face locations. Thus, again using Eq. 21, we have

  $$\begin{aligned} \overline{\frac{AP^{u_{f1}}\,u^{l}_{f1} - \sum \,A_{k}\,u^{l}_{f1,k}}{\Delta V}}\,|_{e} = \overline{R}_{u_{f1, e}} - \overline{\frac{\partial p^{l}}{\partial x_{1}}}\,|_{e} \end{aligned}$$

  (31)

  Effectively, therefore

  $$\begin{aligned} R_{u_{f1, e}} = \overline{R}_{u_{f1, e}} - \overline{\frac{\partial p^{l}}{\partial x_{1}}}\,|_{e} + \frac{\partial p^{l}}{\partial x_{1}}\,|_{e} \end{aligned}$$

  (32)

• 5. Multidimensionally averaged $\overline{R}_{u_{f1, e}}$ is evaluated as

  $$\begin{aligned} \overline{R}_{u_{f1,e}}= & {} \frac{1}{2}\,\left[ \frac{1}{2}\,( R_{u1, P} + R_{u1, E} ) + \frac{\Delta x_{2,n}\,R_{u1,se} + \Delta x_{2,s}\,R_{u1, ne}}{\Delta x_{2,n} + \Delta x_{2,s}} \right] \nonumber \\ R_{u1, se}= & {} \frac{1}{4}\,( R_{u1, P} + R_{u1, E} + R_{u1, S} + R_{u1, SE} ) \nonumber \\ R_{u1, ne}= & {} \frac{1}{4}\,( R_{u1, P} + R_{u1, E} + R_{u1, N} + R_{u1, NE} ) \end{aligned}$$

  (36)

  This representation shows that effectively residuals at nodal locations P, E, N, S, NE and SE only are involved. These residuals will, of course, vanish at full convergence because momentum equations are being solved at the nodal positions. Therefore, effectively $\overline{R}_{u_{f1, e}} = 0$ at convergence and Eq. 35 can be written as

  $$\begin{aligned} R_{u_{f1, e}} = \frac{\partial p^{l}}{\partial x_{1}}\,|_{e} - \overline{\frac{\partial p^{l}}{\partial x_{1}}}\,|_{e} \end{aligned}$$

  (37)

• 6. Now, to evaluate multidimensionally averaged pressure gradient in the above equation, we write

  $$\begin{aligned} \overline{\frac{\partial p^{l}}{\partial x_{1}}}\,|_{e}= & {} \frac{1}{2}\,\left[ \frac{1}{2}\,\left( \frac{\partial p^{l}}{\partial x_{1}}\,|_{P} + \frac{\partial p^{l}}{\partial x_{1}}\,|_{E} \right) \right. \nonumber \\+ & {} \left. \frac{\Delta x_{2, n}\,{\partial p^{l}}/{\partial x_{1}}\,|_{se} + \Delta x_{2, s}\,{\partial p^{l}}/{\partial x_{1}}\,|_{ne}}{\Delta x_{2, n} + \Delta x_{2, s}} \right] \nonumber \\= & {} \frac{1}{4}\,\left[ \frac{p^{l}_{E} - p^{l}_{W}}{\Delta x_{1, e} + \Delta x_{1, w}} + \frac{p^{l}_{EE} - p^{l}_{P}}{\Delta x_{1, e} + \Delta x_{1, w}} \right] \nonumber \\+ & {} \frac{1}{4}\,\frac{\Delta x_{2, s}}{\Delta x_{2, n} + \Delta x_{2, s}}\,\left[ \frac{p^{l}_{E} + p^{l}_{NE} - p^{l}_{P} - p^{l}_{N}}{\Delta x_{1, e}} \right] \nonumber \\+ & {} \frac{1}{4}\,\frac{\Delta x_{2, n}}{\Delta x_{2, n} + \Delta x_{2, s}}\,\left[ \frac{p^{l}_{E} + p^{l}_{SE} - p^{l}_{P} - p^{l}_{S}}{\Delta x_{1, e}} \right] \end{aligned}$$

  (38)

  To simplify the above evaluation further, following definitions are introduced allowing for the non-uniform grid spacing:

  $$\begin{aligned} \overline{p^{l}}_{x_{1}, P}\equiv & {} \frac{\Delta x_{1, w}\,p^{l}_{E} + \Delta x_{1, e}\,p^{l}_{W}}{\Delta x_{1, w} + \Delta x_{1, e}} \text{(solution } \text{ to } \frac{\partial ^{2} p^{l}}{\partial x^{2}_{1}}|_{P} = 0) \end{aligned}$$

  (39)
  $$\begin{aligned} \overline{p^{l}}_{x_{2}, P}\equiv & {} \frac{\Delta x_{2, s}\,p^{l}_{N} + \Delta x_{2, n}\,p^{l}_{S}}{\Delta x_{2, s} + \Delta x_{2, n}} \text{(solution } \text{ to } \frac{\partial ^{2} p^{l}}{\partial x^{2}_{2}}|_{P} = 0) \end{aligned}$$

  (40)
  $$\begin{aligned} \overline{p^{l}}_{P}= & {} \frac{1}{2}\,( \overline{p^{l}}_{x_{1}, P} + \overline{p^{l}}_{x_{2}, P} ) \end{aligned}$$

  (41)
  $$\begin{aligned} \overline{p^{l}}_{x_{1}, E}\equiv & {} \frac{\Delta x_{1, e}\,p^{l}_{EE} + \Delta x_{1, ee}\,p^{l}_{P}}{\Delta x_{1, e} + \Delta x_{1, ee}} \text{(solution } \text{ to } \frac{\partial ^{2} p^{l}}{\partial x^{2}_{1}}|_{E} = 0) \end{aligned}$$

  (42)
  $$\begin{aligned} \overline{p^{l}}_{x_{2}, E}\equiv & {} \frac{\Delta x_{2, s}\,p^{l}_{NE} + \Delta x_{2, n}\,p^{l}_{SE}}{\Delta x_{2, s} + \Delta x_{2, n}} \text{(solution } \text{ to } \frac{\partial ^{2} p^{l}}{\partial x^{2}_{2}}|_{E} = 0) \end{aligned}$$

  (43)
  $$\begin{aligned} \overline{p^{l}}_{E}= & {} \frac{1}{2}\,( \overline{p^{l}}_{x_{1}, E} + \overline{p^{l}}_{x_{2}, E} ) \end{aligned}$$

  (44)

  Substituting the above definitions⁹ in Eq. 38 and replacing $p^{l}_{EE}$ (Eq. 42) and $p^{l}_{W}$ (Eq. 39), respectively, in favour of $p^{l}_{E}$ and $p^{l}_{P}$, it can be shown that

  $$\begin{aligned} \overline{\frac{\partial p^{l}}{\partial x_{1}}}\,|_{e}= & {} \frac{1}{2} \,\left[ \frac{p^{l}_{E} - p^{l}_{P}}{\Delta x_{1, e}} + \frac{\overline{p}^{l}_{E} - \overline{p}^{l}_{P}}{\Delta x_{1, e}}\,\right] \nonumber \\= & {} \frac{1}{2} \,\frac{\partial (\, p^{l} + \overline{p}^{l}\,)}{\partial x_{1}}\,|_{e} \end{aligned}$$

  (45)

  and, therefore, from Eq. 37

  $$\begin{aligned} R_{u_{f1, e}} = \frac{\partial p^{l}}{\partial x_{1}}\,|_{e} - \frac{1}{2} \,\frac{\partial (\, p^{l} + \overline{p}^{l}\,)}{\partial x_{1}}\,|_{e} = \frac{1}{2}\,\frac{\partial ( p^{l} - \overline{p}^{l} )}{\partial x_{1}}\,|_{e} = \frac{\partial p^{'}_{sm}}{\partial x_{1}}\,|_{e} \end{aligned}$$

  (46)

  where

  $$\begin{aligned} p^{'}_{sm} = \frac{1}{2}\,( p^{l} - \overline{p}^{l} ) \; \text{(Smoothing } \text{ Pressure } \text{ Correction) } \end{aligned}$$

  (47)

  Note that the analysis of the discretised equations has yielded a value $\lambda = 0.5$ and $q = p^{'}_{sm}$ (see Eq. 9).

• 7. Repeating items 4, 5 and 6 at other cell faces, it can be shown that

  

  (48)

  Thus, substituting the above equations in Eq. 25, it follows that

  $$\begin{aligned} \dot{m}_{R}= & {} AE\,\frac{\partial p^{'}_{sm}}{\partial x_{1}}\,\Delta x_{1}|_{e} - AW\,\frac{\partial p^{'}_{sm}}{\partial x_{1}}\,\Delta x_{1}|_{w} \nonumber \\+ & {} AN\,\frac{\partial p^{'}_{sm}}{\partial x_{2}}\,\Delta x_{2}|_{n} - AS\,\frac{\partial p^{'}_{sm}}{\partial x_{2}}\,\Delta x_{2}|_{s} \end{aligned}$$

  (49)

• 8. In evaluating coefficients AE, AW, AN and AS, we need AP coefficients at the cell faces (see Eq. 23). But, these can be evaluated by one-dimensional averaging as

  

  (50)

  where the AP coefficients at the nodal locations P, N, E, S, W are known on colocated grids.

• 9. The above derivations show that Eqs. 24 and 25 can be replaced by Eqs. 28 and 49, respectively. Thus, the mass-conserving pressure-correction Eq. 20 appropriate for colocated grids can effectively be written as

  $$\begin{aligned} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l+1}_{m}\,D_{i}\, \frac{\partial p^{'}_{m}}{\partial x_{i}} \right]= & {} \frac{\partial ( \rho ^{l+1}_{m} )}{\partial t} + \frac{\partial ( \rho ^{l+1}_{m}\,\overline{u}^{l}_{i} )}{\partial x_{i}} \nonumber \\- & {} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l+1}_{m}\,D_{i}\, \frac{\partial p^{'}_{sm}}{\partial x_{i}} \right] \end{aligned}$$

  (51)

2.2.1 Further Simplification

It is possible to further simplify Eq. 51. To understand this simplification, consider, for example, the grid disposition near the west boundary as shown in Fig. 5. When computing at the near-boundary node P(2, j), the pressure gradient ${\partial p}/{\partial x_{1}}\,|_{P}$ must be evaluated in the momentum equation for velocity $u_{1, P}$. This will require knowledge of boundary pressure $p_{b} = p\,( 1, j )$. On colocated grids, this pressure is not known and, therefore, is evaluated by linear extrapolation from interior flow points. Thus,

$$\begin{aligned} p_{b} = \frac{L_{bE}}{ L_{PE}}\,p_{P} - \frac{L_{bP}}{L_{PE}}\,p_{E} \end{aligned}$$

(52)

where L denotes length. The same procedure is adopted at Nb and Sb. Now, assuming that the pressure variation near a boundary is locally linear in both $x_{1}$ and $x_{2}$ directions, it follows that

$$\begin{aligned} p_{b} - \overline{p}_{b} = p_{P} - \overline{p}_{P} \text{ or } p^{'}_{sm, b} = p^{'}_{sm, P} \end{aligned}$$

(53)

Fig. 5

West boundary I = 1

and, therefore,

$$\begin{aligned} \frac{\partial p^{'}_{sm}}{\partial x_{1}}\,|_{b} = \frac{\partial p^{'}_{sm}}{\partial n}\,|_{b} = 0 \end{aligned}$$

(54)

The same condition is also applicable to $p^{'}_{m}$ (see Eq. 27). Now, Eq. 51 shows that multipliers of gradients of $p^{'}_{m}$ and $p^{'}_{sm}$ are identical and since the boundary conditions for these two variables are also identical, we may write the mass-conserving pressure correction in the following form:

$$\begin{aligned} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l+1}_{m}\,D_{i}\, \frac{\partial p^{'}}{\partial x_{i}} \right] = \frac{\partial ( \rho ^{l+1}_{m} )}{\partial t} + \frac{\partial ( \rho ^{l+1}_{m}\,\overline{u}^{l}_{i} )}{\partial x_{i}} \end{aligned}$$

(55)

where $p^{'}$ is called total pressure correction and is expressed as

$$\begin{aligned} p^{'} = p^{'}_{m} + p^{'}_{sm} \end{aligned}$$

(56)

Here, $p^{'}_{sm}$, of course, is evaluated from Eq. 47. Following Eqs. 54 and 27, Eq. 55 must be solved with boundary condition

$$\begin{aligned} \frac{\partial p^{'}}{\partial x_{1}}\,|_{b} = \frac{\partial p^{'}}{\partial n}\,|_{b} = 0 \end{aligned}$$

(57)

The discretised form of Eq. 55 is

$$\begin{aligned} AP\,p^{'}_{P} = AE\,p^{'}_{E} + AW\,p^{'}_{W} + AN\,p^{'}_{N} + AS\,p^{'}_{s} - \overline{\dot{m}}_{P} \end{aligned}$$

(58)

where $\overline{\dot{m}}_{P}$ is given by Eq. 28.

Finally, note that Eq. 55 has similarity with the staggered grid equation (26). Both are Poisson’s equations with similar boundary conditions (57) and (27).

2.2.2 Modification for Compressible Flow

So far it has been assumed that $\rho _{m}$ = constant as in incompressible flow. However, in compressible flow $\rho _{m}$ is connected to pressure p via equation of state such as $p = \rho _{m}\,R_{g}\,T$. Thus,

$$\begin{aligned} \rho ^{l+1}_{m} = \rho ^{l}_{m} + \rho ^{'}_{m} = \rho ^{l}_{m} + \frac{p^{'}_{m}}{R_{g}\,T} = \rho ^{l}_{m} + \frac{p^{'} - p^{'}_{sm}}{R_{g}\,T} \end{aligned}$$

(59)

With this substitution, Eq. 55 will read as

$$\begin{aligned} \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l}_{m}\,D_{i}\, \frac{\partial p^{'}}{\partial x_{i}} - U^{*}_{i}\,\frac{p^{'}}{R_{g}T} \right]= & {} \frac{\partial ( \rho ^{l}_{m} )}{\partial t} + \frac{\partial }{\partial x_{i}}\,\left[ \rho ^{l}_{m}\,\overline{u}^{l}_{i} - U^{*}_{i}\,\frac{p^{'}_{sm}}{R_{g}T} \right] \end{aligned}$$

(60)

$$\begin{aligned} U^{*}_{i}= & {} u^{l}_{i} - D_{i}\,\frac{\partial p^{'}_{sm}}{\partial x_{i}} \end{aligned}$$

(61)

The left-hand side of Eq. 60 contains diffusion ($D_{i}{\partial p^{'}}/{\partial x_{i}}$) as well as convection ($U^{*}_{i}\, p^{'}/R_{g}T$) terms. Thus, for compressible flow, the pressure-correction equation is a transport equation for $p^{'}$ and not a Poisson’s equation. Also, note that if $U^{*}_{i} = 0$, the incompressible form (see Eq. 51) is recovered.

2.2.3 Overall Calculation Procedure on Colocated Grids

The sequence of calculations on colocated grids is as follows:

• 1. At a given time step, guess pressure field $p^{l}$. This may be the pressure field from the previous time step.

• 2. Solve momentum equations once for each $u_{i}$ with problem-dependent boundary conditions. Thus

  $$\begin{aligned} AP^{u_{i}}\,u^{l}_{i,P} = \sum \,A^{u_{i}}_{k}\,u^{l}_{i,k} - \Delta V\,\frac{\partial p^{l}}{\partial x_{i}}|_{P} + \frac{\rho ^{o}_{m,p}\,\Delta V}{\Delta t}\,u^{o}_{i,P} \end{aligned}$$

  (62)

  where $AP^{u_{i}} = \sum \,A^{u_{i}}_{k} + \rho ^{0}_{m}\,\Delta V/\Delta t$.

• 3. Using the $u^{l}_{i}$ distribution, solve Eq. 55 (or, 60) with boundary condition (57) to yield the total pressure correction $p^{'}_{i, j}$ field. This implies iterative solution of Eq. 58. The number of iterations typically may not exceed 10.

• 4. Recover mass-conserving pressure correction via Eq. 56. Thus

  $$\begin{aligned} p^{'}_{m,P}= p^{'}_{P} - p^{'}_{sm,P} = p^{'}_{P} - \frac{1}{2}\,( p^{l}_{P} - \overline{p}^{l}_{P} ) \end{aligned}$$

  (63)

  where $\overline{p}^{l}_{P}$ is evaluated from Eqs. 15 and 16. Using this $p^{'}_{m}$ field, the mass conservation error is evaluated from¹⁰

  $$\begin{aligned} R_{\text {mass},P} = AP\,p^{'}_{m,P} -( AEp^{'}_{m,E} + AWp^{'}_{m,W} + ANp^{'}_{m,N} + ASp^{'}_{m,S}) \end{aligned}$$

  (64)

• 5. Correct pressure and velocity fields according to

  

  (65)
  $$\begin{aligned} u^{l+1}_{i,P} = u^{l}_{i,P} - D_{i}\, \frac{\partial p^{'}_{m}}{\partial x_{i}}|_{P} \end{aligned}$$

  (66)

• 6. Using this new velocity field, solve discretised forms of scalar transport equations relevant to the problem at hand.

• 7. Evaluate residuals $R_{\Phi }$ of momentum and scalar ($\Phi$) equations. The mass residual $R_{m}$ evaluated from Eq. 64. When maximum residual as per the L2-norm is ${<} 10^{-5}$, convergence is declared.

• 8. If this convergence criterion is not satisfied, treat $p^{l+1} = p^{l}$, $u^{l+1}_{i} = u^{l}_{i}$ and $\Phi ^{l+1} = \Phi ^{l}$ and return to step 2.

• 9. To execute the next time step, set $u^{o}_{i} = u^{l + 1}_{i}$, $\Phi ^{o} = \Phi ^{l + 1}$ and return to step 1.

2.3 Some Illustrative Problems

In all problems, computations are carried out with ($\lambda = 0.5$) and without ($\lambda = 0$) application of smoothing pressure correction.

2.3.1 Essentially Parabolic Flows

We consider 2D laminar developing flow between two parallel plates 2b apart ($D_{h} = 4b$). Although the flow is parabolic, here it is treated as being governed by 2D elliptic equations.

Fig. 6

Entrance region flow between parallel plates

Figure 6a shows predicted axial variation of wall pressure for the case when the fluid enters the channel with uniform axial velocity $u_{\text {in}} = \overline{u}$. As the boundary layers develop along the channel walls, the true pressure variation is expected to be non-linear with x in the entrance region but, at large x, it is expected to be linear because the flow is now fully developed. It is seen that with $\lambda =0$, the predicted pressure is zig-zag near the entrance, whereas at large x, pressure zig-zagness disappears.

In contrast, Fig. 6b shows predictions for the case in which the fluid enters with fully developed parabolic axial velocity profile. Then, it is expected that true axial pressure variation will be linear right from the $x=0$. Figure 6b confirms this expectation even when $\lambda = 0$.

Fig. 7

Natural convection in horizontal concentric annulus [4]

2.3.2 Essentially Elliptic Flows

Natural Convection: Figure 7 (left) shows concentric cylinders in which the inner cylinder (dia $D_{i}$) is hotter ($T_{h}$) than the outer ($T_{c}$) one (dia $D_{o}$). The annulus gap width $L =(D_{o} - D_{i})/2$. Figure 7 (right) shows predicted pressure variation along the horizontal plane in the presence of natural convection. Here again, since the true pressure variation is non-linear, the predicted pressure with $\lambda = 0$ shows zig-zagness. But, with smoothing pressure correction ($\lambda = 0.5$), zig-zagness disappears. The annulus natural convection flow is fully elliptic that involves simultaneous solution of the energy equation along with the flow equations. Further, the domain of computation is mapped by unstructured grid of triangular elements [4].

Mixed Convection: We now consider the problem of mixed convection flow in a right-angled corner. As shown in Fig. 8 (top), flow enters the domain of computation at the north boundary, turns anticlockwise around the corner and leaves through the east boundary. For this problem, an exact solution is developed by Shih and Ren [28] by specifying artificial boundary conditions. The unique feature of the exact solution is that the pressure variation in Y-direction is linear at all constant X-planes. Figure 8 (left) shows comparison of predicted and exact profiles of velocity (u and v). Similar comparison for pressure is shown in Fig. 8 (right). It is seen that since the exact pressure variation is linear in Y, no zig-zagness is predicted for both $\lambda = 0$ and 0.5.

Fig. 8

Mixed convection in a corner—artificial boundary conditions

2.3.3 Compressible Flow in 2D Plane Nozzle

Figure 9 (top) shows the computational domain of a 2D convergent-divergent plane nozzle. The bottom boundary represents the axis (centreline) of the nozzle whereas the top boundary is a wall. The flow enters the left boundary and leaves through the right boundary. The total nozzle length $L = 11.56$ cm and the throat is midway. The half-heights of the nozzle at entry, throat and exit are 3.52 cm, 1.37 cm and 2.46 cm, respectively. The inlet Mach number is $M_{\text {in}} = 0.232$ and the exit static pressure is $p\,/\,p_{0} = 0.1135$ where $p_{0}$ is the stagnation pressure. The stagnation enthalpy is assumed constant. For these specifications, experimental data are available [16]. This flow has been computed by Karki and Patankar [13] using curvilinear grids and Upwind Difference Scheme (UDS) for the convective terms and assuming $\mu = 0$ (that is, Euler equations are solved). Here, the flow is computed using unstructured mesh and Total Variation Diminishing (TVD) scheme [14, 17] again with $\mu = 0$.

Fig. 9

2D plane nozzle: computational domain (top), variation of p and M (middle), Mach number contours 0.2 (0.1) 2.0 (bottom)

At inflow plane, since $M_{\text {in}}$ is known, $u_{\text {in}}$, $T_{\text {in}}$ and $p_{\text {in}}$ are specified using standard isentropic relationships [10]. In the exit plane, except for pressure (which is fixed), all other variables are linearly extrapolated from the near-exit boundary node values. At the upper nozzle wall, tangency condition is applied. The pressure distribution is determined by discretising a compressible flow version of the total pressure-correction equation (60). For velocities, equations for $u_{1}, u_{2}$ are solved and temperature is recovered from definition of stagnation enthalpy. Finally, density is determined using equation of state $p = \rho _{m}\,R_{g}\,T$. Computations are performed using 570 elements.

Figure 9 (middle) shows the predicted variations of pressure (dashed line) and Mach numbers (solid line) at the upper wall and the centreline. The experimental data (open circles) for pressure have been read from a figure in [13]. It is seen that the agreement between experiment and predictions is satisfactory. Note that the predicted Mach number at the upper wall passes through $M = 1$ exactly at the throat ($X/L = 0.5$) and reaches supersonic state $M = 2.01$ at exit. At the centreline, however, $M = 1$ location is downstream of the throat. Finally, Fig. 9 (bottom) shows the iso-Mach contours. Notice that the iso-Mach lines are slanted which have been found to be in agreement with computations of [13].

2.4 Main Findings

The above examples confirm our theoretical deductions (see Sect. 1.2) on importance of including simple algebraic smoothing pressure correction $p^{'}_{sm}$ on colocated grids to avoid prediction of zig-zag pressure. The efficacy of the smoothing correction has been shown in respect of

• 1. Incompressible as well as compressible flows.

• 2. Problems governed by parabolic and elliptic equations.

• 3. Problems with/without body forces.

• 4. Structured as well as unstructured grids.

• 5. Applicability to any boundary condition.

Several other applications of more complex flows can be found in [7, 11, 22, 23, 30] by way of example.

3 Interfacial Flows—Volume Error

3.1 Single-Fluid Formalism

In interfacial unsteady flows of two immiscible fluids, a problem of loss of fluid volume/mass is encountered when computing in discretised space. This problem is particularly severe when the density ratio of the two fluids is large (as, for example, in air and water). Such interfacial problems are often solved by employing single-fluid formalism in which the flow of two fluids is treated as that of a single fluid whose properties change abruptly across the interface. In such flows, the governing equations are [17]

$$\begin{aligned} \frac{\partial u_{f,j}}{\partial x_{j}}= & {} \bigtriangledown \,.\,\vec {V}_{f} = 0 \; \text{(Volume } \text{ Conservation) } \end{aligned}$$

(67)

$$\begin{aligned} \frac{\partial (\rho _{m}\,u_{i})}{\partial t}+ & {} \frac{\partial }{\partial x_{j}}\,(\rho _{m}\,u_{f,j}\,u_{i}) = \frac{\partial }{\partial x_{j}}\,\left[ \mu _{m}\, \frac{\partial u_{i}}{\partial x_{j}} \right] - \frac{\partial p}{\partial x_{i}} \nonumber \\+ & {} \rho _{m}\, g_{i} + F_{st,i} + \frac{\partial }{\partial x_{j}}\,\left[ \mu _{m}\, \frac{\partial u_{j}}{\partial x_{i}} \right] \end{aligned}$$

(68)

$$\begin{aligned} \frac{\partial \rho _{m}}{\partial t}+ & {} \frac{\partial }{\partial x_{j}}( \rho _{m}\,u_{f,j}) = 0 \;\text{(Mass } \text{ Conservation) } \end{aligned}$$

(69)

The above equations are special in that the pressure is determined from the continuity equations (67) (or volume conservation equation) but the superficial density $\rho _{m}$ is determined from the conserved scalar equation (69) (or mass conservation equation). Thus, the equations carry characteristics of both an incompressible flow and a compressible flow. As such, incorporation of $q_{\text {disc}}$ becomes important. Then, following from Eq. 10 and combining Eqs. 67 and 69, it can be shown that

$$\begin{aligned} p^{'}_{sm}= & {} q_{\text {disc}} = \frac{1}{2}\,( p - \overline{p} ) + \gamma \,\bigtriangledown \,.\,V_{f} \nonumber \\= & {} \frac{1}{2}\,( p - \overline{p} ) - \frac{\gamma }{\rho _{m}}\,\left\{ \frac{\partial \rho _{m}}{\partial t} + u_{f,j}\, \frac{\partial \rho _{m}}{\partial x_{j}} \right\} \nonumber \\= & {} \frac{1}{2}\,( p - \overline{p} ) - \gamma \, \frac{D}{Dt} (\text{ ln }\,\rho _{m} ) \end{aligned}$$

(70)

Notice that the last term will become significant (in the neighbourhood of the interface in discretised CFD) when density ratio of the two fluids is large. Most authors (see, for example [9, 25, 31,32,33]) instead of solving Eq. 69, solve an equation for Volume Fraction F of the heavier fluid a (say). F is defined as

$$\begin{aligned} F \equiv \frac{\rho _{m} - \rho _{b}}{\rho _{a} - \rho _{b}} \end{aligned}$$

(71)

where suffix b stands for the lighter fluid.¹¹ The interface is now notionally identified with $F = 0.5$. The predicted smeared (about $F = 0.5$) F-distribution at a time step is then corrected to conserve volume in the two-fluid CVs (in which $0< F < 1.0$ as shown in Fig. 13-left) on the basis of geometric considerations. The correction procedures become extremely complex in 3D flows [25]. Other authors identify the interface with Level Set (usually zero) and define a level-set distance function [31]. The level-set equation is same as the F-equation but the correction procedure is invoked such that $|\bigtriangledown \, \cdot \, F|$ = 1. In both procedures, apart from the additional work requirement, volume/mass balance errors arise [33].

Use of smoothing pressure correction indicated in Eq. 70, on the other hand, is very simple to implement and avoids volume/mass errors (within discretisation errors). To illustrate this, we consider few problems¹² with and without effects of surface tension.

Fig. 10

Splashing of a water drop on a surface: $F=0.5$ contours (left), pressure and velocity vectors (right)

3.2 Problems with $F_{st}$ = 0

3.2.1 Splashing of a Drop on a Liquid Surface

Consider a two-dimensional rectangular enclosure (Fig. 10) of 7 mm $\times$ 14 mm dimensions. The enclosure is filled with water to a height of 8.75 mm. Initially, a cylindrical water drop of radius $r_{d} = 1.4$ mm is placed above the water surface in air at the centre plane $x_{1} =$ 3.5 mm and height $x_{2} =$ 10.55 mm. For t > 0, the drop falls under the action of gravity (g = 9.81 m/$\text {s}^{2}$) and splashes on the water surface creating ripples and merges with the body of water.

Computations are performed with 64 $\times$ 128 grid without exploiting symmetry. 50 iterations per time step ($10^{-5}$ s) are required with $p^{'}$ under-relaxed for obtaining convergence. Computations are continued up to $t = 0.025$ s and the interface profiles ($F=0.5$) are shown in Fig. 10 (left). The present results had maximum volume error¹³ 0.002% at the last time step used in the computations. Notice also the smooth pressure prediction and the velocity vectors in Fig. 10 (right). Velocities in air have higher magnitude than in water. These motions cause development of pressure variations that deviate from pure hydrostatic pressure variation.

Fig. 11

Collapse of a water column

3.2.2 Collapse of a Water Column

Figure 11 (top) shows the problem configuration and the domain of computation. Initially, water (fluid a) column $H = 2$ m high and $L = 1$ m wide is kept at rest by means of a dam. Fluid b is air. At $t=0$, the dam breaks resulting in collapse of water column followed by horizontal spread of water. Computations are performed with $40 (x_{1}) \times 22 (x_{2})$ uniform grid. Taking $g = 9.81$, fixed time step $\Delta \tau = 0.001$ is used where $\tau = t \times ( 2\times g / L )^{0.5}$. Computations are carried out till $\tau = 3$. At this time, the maximum error in volume balance was found to be 0.432%.

Figure 11 (middle) shows the $F = 0.5$ contours at $\tau$ = 0.9, 1.4, 2.0 and 3.0. These contours mimic those computed by Jun and Spalding [12] using the explicit van Leer scheme [35]. Thus, the TVD scheme [14] used in present computations succeeds in sharp interface capturing. The time variations of horizontal spread X and of vertical fall Y (solid lines) are compared in Fig. 11 (bottom) with experimental data (open circles) of Martin and Moyce [15]. The comparison is reasonable notwithstanding the experimental difficulties mentioned by them.

Fig. 12

Sloshing in a tank (top), interface ($F=0.5$) locations (middle), pressure contours and velocity vectors (bottom) arrow size: 1 cm = 13 cm/s

3.2.3 Sloshing in a Tank

Figure 12 (top) shows the problem specification. The tank ($40 \times 25$ cm) is moved with a horizontal displacement $x_{1}\,( t ) = A\,\left\{ \text{ sin }\,( 2\,\pi \,f_{1}\,t ) - \text{ sin }\,( 2\,\pi \,f_{2}\,t ) \right\}$ where $A = 7.5 \times 10^{-3}$, $f_{1} = 1.598$ Hz and $f_{2} = 1.307$ Hz. Computations are performed with $60 \times 60$ cells and time step $\Delta t = 0.001$ s. On all boundaries, $u_{1} ( t ) = \text {d}x_{1}/\text {d}t, u_{2} = 0$ was specified at each time step. Nearly 500 iterations per step are required to obtain convergence. The pressure-correction equation was under-relaxed in the first 100 time steps to procure convergence. At $t = 2.004$ s, the volume error (see Eq. 72) was 0.45%.

In Fig. 12 (middle), the predicted interface profiles (lines) are compared with experimental data (dots) as read from paper by Andrillon and Alessandrini [1]. Again, the agreement is seen to be reasonable. Figure 12 (bottom) shows dimensionless pressure $p^{*} = ( p - p_{\text {min}} ) / ( p_{\text {max}} - p_{\text {min}} )$ contours and velocity vectors at different times. The velocity in the air greatly exceeds that in the water. The fluid re-circulations due to interface movement accord with the expectation and the pressure contours are indeed smooth.

3.3 Problems with Finite $F_{st}$

In computation of many interfacial flows involving merger and splits of the interfaces, the surface tension force $F_{st}$ is included in the momentum equations. The force acts tangent to the interface. However, the interface within a control volume is taken to be locally spherical. As such, the net force acts normal to the interface while the net forces normal to the interface normal cancel out. Thus, the net force per unit volume in direction i is given by [17]

$$\begin{aligned} F_{st, i}= & {} -\,\sigma \,\kappa \,\frac{\partial F}{\partial x_{i}} = -\,\sigma \,\kappa \,\frac{\partial F^{*}}{\partial x_{i}} \end{aligned}$$

(73)

$$\begin{aligned} \text{ where } F^{*}\equiv & {} 0.5\,\left\{ 1 + \frac{( F - 0.5 )}{ | ( F - 0.5 ) |} \right\} \end{aligned}$$

(74)

where F is the volume fraction of the heavier fluid a and $\kappa$ is the interface curvature. The replacement of F by $F^{*}$ simply ensures that the surface tension force is evaluated at the interface ($F = 0.5$) only even when the F-distribution is smeared. Of course, $F^{*}_{b} = 0$ and $F^{*}_{a} = 1$.

3.3.1 Geometric Evaluation of $\kappa$

Most authors using VOF or LS methods, evaluate $\kappa$ from reconstructed F-distribution as

$$\begin{aligned} \kappa= & {} -\,\frac{1}{A}\,\left[ \frac{1}{A}\,\frac{\partial F}{\partial x_{i}}\,\frac{\partial A}{\partial x_{i}} - \frac{\partial ^{2} F}{\partial x^{2}_{i}} \right] \text{ summation } \end{aligned}$$

(75)

$$\begin{aligned} \text{ where }A= & {} \sqrt{( \frac{\partial F}{\partial x_{1}})^{2} + ( \frac{\partial F}{\partial x_{2}})^{2} + ( \frac{\partial F}{\partial x_{3}})^{2}} \end{aligned}$$

(76)

Following comments are now considered pertinent:

• 1. Evaluation of $\kappa$ according to Eq. 75 is complex and is known to introduce discretisation errors. This has been shown by Takahira et al. [33] where a computation of a static bubble surrounded by static liquid generates spurious velocities.

• 2. Further, it is important to point out that many authors [2, 9, 26] study effect of surface tension coefficient $\sigma$ by keeping the density and viscosity values of two fluids unchanged.¹⁴ As such, $F_{st, i}$ calculated using Eq. 75 produces different magnitudes of the force (see Eq. 73) for the same fluid pair. However, in the literature, no real two fluid pairs having same values of density and viscosity but different values of $\sigma$ are found (see, for example [21]).

• 3. The source of the difficulty mentioned above, however, can be traced to non-dimensionalisation of momentum equation (68). Many authors (see [33], for example) use reference velocity U, reference length L and reference properties $\rho _{a}$ and $\mu _{a}$ to non-dimensionalise equation (68). The dimensionless equation then reads

  $$\begin{aligned} \frac{\partial \rho ^{*}_{m}\,u^{*}_{i}}{\partial t^{*}}+ & {} \frac{\partial \rho ^{*}_{m}\,u^{*}_{f,j}\,u^{*}_{i}}{\partial x^{*}_{j}} = \frac{1}{Re}\,\frac{\partial }{\partial x^{*}_{j}}\,\left[ \mu ^{*}_{m}\, \frac{\partial u^{*}_{i}}{\partial x^{*}_{j}} \right] - \frac{\partial p^{*}}{\partial x^{*}_{i}} \nonumber \\+ & {} \rho ^{*}_{m}\, g^{*}_{i} - \frac{\kappa ^{*}}{We}\,\frac{\partial F^{*}}{\partial x^{*}_{i}} + \frac{1}{Re}\, \frac{\partial }{\partial x^{*}_{j}}\,\left[ \mu ^{*}_{m}\, \frac{\partial u^{*}_{j}}{\partial x^{*}_{i}} \right] \end{aligned}$$

  (77)

  where the dimensionless terms are

  $$\begin{aligned} Re= & {} \frac{\rho _{a}\,U\,L}{\mu _{a}} \text{ Reynolds } \text{ number } \end{aligned}$$

  (78)
  $$\begin{aligned} g^{*}= & {} \frac{g\,L}{U^{2}} \text{ Froude } \text{ number } \end{aligned}$$

  (79)
  $$\begin{aligned} \kappa ^{*}= & {} \kappa \,L \text{ Dimensionless } \text{ Curvature } \end{aligned}$$

  (80)
  $$\begin{aligned} We= & {} \frac{\rho _{a}\,U^{2}\,L}{\sigma } \text{ Weber } \text{ number } \end{aligned}$$

  (81)
  $$\begin{aligned} \rho ^{*}_{m}, \mu ^{*}_{m}= & {} \frac{\rho _{m}}{\rho _{a}}, \frac{\mu _{m}}{\mu _{a}} \text{ Dimensionless } \text{ properties } \end{aligned}$$

  (82)
  $$\begin{aligned} u^{*}_{i}, p^{*}= & {} \frac{u_{i}}{U},\, \frac{p}{\rho _{a}\,U^{2}} \text{ Dimensionless } \text{ Velocity } \text{ and } \text{ Pressure } \end{aligned}$$

  (83)
  $$\begin{aligned} x^{*}_{i}, t^{*}= & {} \frac{x_{i}}{L}, \,\frac{t}{L / U} \text{ Dimensionless } \text{ coordinates } \text{ and } \text{ Time } \end{aligned}$$

  (84)

  Equation 77 thus shows that since $\kappa ^{*}$ is evaluated geometrically from F-distribution, the Weber number (We) now appears to be an independent parameter of the flow system. This interpretation leads to investigation of effect of surface tension coefficient $\sigma$ (or We) for the same fluid pair. In the discussion below, we show that this is misleading and that Weber number is not an independent parameter of the flow system.

Fig. 13

Illustrative F-distribution and interface normal (left), zero-thickness CV surrounding an interface (right)

3.3.2 Fluid Dynamic Evaluation of $\kappa$

In view of the last comment, we consider an alternative approach to evaluation of $\kappa$ and $F_{st, i}$. Thus, we assume that the interface is a surface of zero thickness having no physical properties as shown in Fig. 13 (right). Then, taking the dot product of momentum equation (68) (along with Eq. 73) with interface normal $\vec {n}$ will result in

$$\begin{aligned} \frac{\partial }{\partial n}\,( p - \tau _{nn} ) = -\sigma \,\kappa \,\frac{\partial F^{*}}{\partial n} \rightarrow \tau _{nn} = 2\,\mu \,\frac{\partial V_{n}}{\partial n} \end{aligned}$$

(85)

and $V_{n}$ is the velocity normal to the interface. Note that the unsteady, convective and gravity terms disappear because the interface has zero thickness and no mass. However, invoking the Stokes’s continuum requirement (with $q = 0$ for a control volume of zero thickness normal to the interface), we have

$$\begin{aligned} p - \tau _{nn} = \overline{p} = -\,\sigma _{n} \end{aligned}$$

(86)

where $\sigma _{n}$ is total normal stress. Hence, Eq. 85 will read as

$$\begin{aligned} \frac{\text {d}\, \overline{p} }{\text {d}\, n} = -\sigma \,\kappa \,\frac{\text {d}\, F^{*}}{\text {d}\, n} \end{aligned}$$

(87)

This equation is same as the familiar Young–Laplace equation for the equilibrium of a static bubble in a static fluid in which $\tau _{nn} = 0$ [29]. Here, $\tau _{nn}$ is finite. Now, upon discretisation along the normal to the interface, it follows that (see Fig. 13 (right))

$$\begin{aligned} \sigma \,\kappa = -\,\frac{\text {d}\,\overline{p} / \text {d}\,n}{\text {d}\,F^{*} / \text {d}\,n} = -\,\frac{\overline{p}_{b} - \overline{p}_{a}}{ F^{*}_{b} - F^{*}_{a}} = -\,\frac{\overline{p}_{b} - \overline{p}_{a}}{ 0 - 1} = \overline{p}_{b} - \overline{p}_{a} \end{aligned}$$

(88)

and hence

$$\begin{aligned} F_{st, i} = -\,( \overline{p}_{b} - \overline{p}_{a} )\,\frac{\partial F}{\partial x_{i}} \end{aligned}$$

(89)

This Fluid Dynamic expression with mean pressure difference ($\overline{p}_{b} - \overline{p}_{a}$) on either side of the interface is again a new result. On both structured and unstructured grids, the difference of average pressures can be evaluated in the following Eq. 15:

$$\begin{aligned} \sigma \,\kappa= & {} ( \overline{p}_{b} - \overline{p}_{a} )_{P} = \frac{1}{3}\,\sum _{i=1}^{3}( \overline{p}_{b} - \overline{p}_{a} )_{x_{i}, P} \end{aligned}$$

(90)

$$\begin{aligned} ( \overline{p}_{b} - \overline{p}_{a} )_{x_{i}, P}= & {} \text{ solution } \text{ of } \frac{\partial ^{2} }{\partial x^{2}_{i}}\,\left[ p\,( 1 - 2\,F^{*} ) \right] _{P} = 0 \end{aligned}$$

(91)

This manner of evaluation ensures that $\sigma \,\kappa$ is calculated at the interface ($F = 0.5$) only.

Thus, Eq. 89 evaluates the surface tension force directly from flow variables without evaluating the interface curvature. As such, the remarkable feature of this expression is that it does not require knowledge of surface tension coefficient $\sigma$.

Fig. 14

Bursting of a bubble

3.3.3 Bursting of an Air Bubble Through Water

To demonstrate validity of two different methods of evaluating the surface tension force, we consider the bubble-burst problem (see Fig. 14). This problem has been solved by Takahira et al. [33] using the level-set method using $60 \times 60 \times 120$ grid and the surface tension force is evaluated from geometric considerations (which requires knowledge of the value of $\sigma$). Here, in order to save computer time, the same problem is solved on coarser grid of dimensions $30 \times 30 \times 60$. Maximum of 50 iterations per time step ($\Delta \tau = 0.001$) are required to procure convergence. The reference velocity $U_{\text {ref}}$, properties $\rho _{a}, \mu _{a}$ and surface tension coefficient $\sigma$ are chosen such that

$$Re = \frac{\rho _{a}\,U_{\text {ref}}\,D}{\mu _{a}} = 474 \qquad We = \frac{\rho _{a}\,U^{2}_{\text {ref}}\,D}{\sigma } = 1.0 $$

$$ Fr = \frac{U^{2}_{\text {ref}}}{g\,D} = 0.64 \frac{\rho _{b}}{\rho _{a}} = 0.001 \frac{\mu _{b}}{\mu _{a}} = 0.01 $$

The problem has been computed in three different ways as follows:

• 1. $F_{st}$ from Eqs. 73, 75 and 76 $\rightarrow$ Fig. 15a.

• 2. $F_{st}$ from Eqs. 89, 90 and 91 $\rightarrow$ Fig. 15b.

• 3. Ignoring surface tension force. That is, $F_{st} = 0$ $\rightarrow$ Fig. 15c.

Figure 15a, b show that predictions with both types of evaluations of $F_{st}$ are nearly identical. The predictions also accord with the fine grid solutions obtained by [33] although the burst heights at large times are somewhat smaller due to coarseness of the grid used here. Nonetheless the formation of ripples on the liquid surface is clearly seen at $\tau$ = 0.4 and 0.72. Likewise, at $\tau$ = 1.4 and 1.68, formation of a neck in the entrained liquid suggests that a detached liquid drop is about to form.

Fig. 15

Interface evolutions during bursting: a geometric evaluation of $F_{st}$, b fluid dynamic evaluation of $F_{st}$, c $F_{st}=0$

Since predictions with both types of evaluations are nearly identical, computations are repeated with $F_{st} = 0$ (see Fig. 15c). It is seen that the burst heights are now greater indicating that the absence of surface tension force fails to minimise the interface surface during bursting, as expected.

Finally, the time variation of volume error (see Eq. 72) evaluated on coarse grid is found to increase with time in an oscillatory manner [18]. At $\tau = 1.68$, the maximum error with fluid dynamic evaluation of $F_{st}$ is 0.078%. The same was found to be 0.093% with geometric evaluation. These errors are much smaller than the ones reported by Takahira et al. [33] on a finer grid.

Incidentally, solution to the problem of merger of two asymmetrically placed air bubbles rising in a box of square cross-section filled with water has been reported in [18]. These solutions further support the deduction that Weber number is not an independent parameter of the system.

4 Conclusions

• 1. Continuum requirement of Stokes’s stress laws is obeyed to

  • a. eliminate problem of zig-zagness pressure prediction in computation of incompressible flow on colocated grids.

  • b. eliminate (within discretisation errors) the problem of loss of mass/volume at large times encountered in computation of unsteady interfacial flows.

  • c. evaluate surface tension force in interfacial flows using dynamic flow variables and without requiring knowledge of surface tension coefficient $\sigma$.

• 2. Thermodynamic explanation for validity of Stokes’s continuum requirement is presented with new interpretations that follow from its violations.