Drag with external and pressure drop with internal flows: a new and unifying look at losses in the flow field based on the second law of thermodynamics

In fluid mechanics external and internal flows are commonly treated as flows that belong to two different categories. When losses are determined for these flows drag is the target quantity for external flows, whereas internal flows are characterized by their pressure drop. In a non-dimensional form they are expressed as drag coefficient c_D and friction factor f, which are

$$\begin{equation} {c_{\rm D}} \equiv \frac{D}{\frac{\varrho}{2} u_\infty^2 A} \quad {\rm (drag~coefficient,~external~flow),} \end{equation} \tag{ 1 }$$

$$\begin{equation} {f}\equiv \frac{(-{\rm d}p/{\rm d}x){D_{\rm h}}}{\frac{\varrho}{2} u_{\rm m}^2 } \quad {\rm~(friction~factor,~internal~flow).} \end{equation} \tag{ 2 }$$

Here, D is the drag force on the body, dp/dx is the pressure drop in a conduit, u_∞ is the undisturbed oncoming flow and u_m is the cross section averaged velocity. In order to end up with a non-dimensional c_D and f, the projected area A and the hydraulic diameter D_h are included.

Figure 1 shows qualitatively how c_D and f look. Here, a Newtonian fluid flows through the pipe (internal flow, characterized by f) and around it (external flow, characterized by c_D) as one would anticipate in a shell-and-tube heat exchanger. Both diagrams look roughly similar though in both cases exactly the same physical situation prevails: mechanical energy is converted into internal energy by a dissipation process (accompanied by entropy generation). Therefore, it should be possible to treat both cases consistently.

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As an alternative to the conventional definitions c_D and f for the drag coefficient and the friction factor, respectively, we suggest to consistently use the entropy generation due to the dissipation of mechanical energy for defining loss coefficients.

For external flows the overall entropy generation rate due to dissipation, $\dot S_{\rm D}$, can be linked to the drag D by considering the (lost) mechanical power P_L:

$$\begin{equation} P_{\rm L} = D u_\infty, \end{equation} \tag{ 3 }$$

which is a dissipation rate, linked to the entropy generation using the ambient temperature T:

$$\begin{equation} P_{\rm L} = T \dot S_{\rm D}. \end{equation} \tag{ 4 }$$

With $D=\dot S_{\rm D} T /u_\infty$ from (3) and (4) the drag coefficient c_D now is, see equation (1),

$$\begin{equation} {c_{\rm D}} =\frac{T}{\frac{\varrho}{2} u_\infty^3 A} \dot S_{\rm D}, \end{equation} \tag{ 5 }$$

For internal flows the pressure drop (− dp/dx) as introduced in equation (2) is a measure for the loss of mechanical energy due to dissipation only for some special (but often occurring) cases. Only when the flow is fully developed and horizontal does (− dp/dx) correspond to the (cross section averaged) dissipation rate. A fully developed flow can occur in pipes and channels only, so that f is no general loss coefficient for internal flows, see Herwig and Schmandt (2012). Its relation to the general head loss coefficient K for internal flows is

$$\begin{equation} {K}={f}\frac{L}{{D_{\rm h}}}, \end{equation} \tag{ 6 }$$

since losses obviously are ∝L and ∝1/D_h in pipes and channels.

The lost mechanical power for internal flow again is due to the dissipation process involved, so that a general definition of K is

$$\begin{equation} {K}=\frac{\varphi}{u^2_{\rm m}/2} \end{equation} \tag{ 7 }$$

with φ as the specific dissipation rate in the device in which the internal flow occurs. This dissipation rate is linked to the entropy generation by

$$\begin{equation} \varphi=\frac{T}{\dot m} \dot S_{\rm D} \end{equation} \tag{ 8 }$$

so that (7) now becomes

$$\begin{equation} {K}=\frac{T}{\frac{u^2_{\rm m}}{2} \dot m} \dot S_{\rm D}. \end{equation} \tag{ 9 }$$

Comparing c_D, see equation (5), and this K it turns out that both loss coefficients now are very similar. With $\dot m = \varrho {u_{\mathrm { m}}} A_{\mathrm { m}}$ this similarity is even stronger, as can be seen in table 1.

Table 1. Drag and head loss coefficients defined consistently with the entropy generation rate $\dot S_{\rm D}$.

External flow	Internal flow
$\displaystyle {c_{\rm D}} = \frac {T}{\frac {\varrho }{2} u_\infty ^3 A} \dot S_{\rm D}$	$\displaystyle {K}= \frac {T}{\frac {\varrho }{2} {u_{\mathrm { m}}}^3 A_{\mathrm { m}}} \dot S_{\rm D}$
(drag coefficient)	(head loss coefficient)
A: projected area	Am: cross section
u∞: oncoming flow velocity	um: cross section averaged velocity

Even though the areas A and A_m and the velocities u_∞ and u_m are different in character the combination ϱu³A has a common meaning. It is the real exergy flux for internal flows and for external flows an exergy flux through a fictitious stream tube with the cross section A.

An important aspect of the definitions in table 1 is the occurrence of the temperature T. For isothermal situations at ambient temperature, T is equal to the ambient temperature T_∞.

When heat transfer is involved T must be taken as a mean temperature T_m within the flow field. If this T_m is equal to the ambient temperature T_∞, c_D and K immediately quantify the loss of exergy. If, however, T_m is not equal to T_∞ these coefficients are still measure of the dissipation of mechanical energy but no longer a measure of the lost exergy. Since in general according to the so-called Gouy–Stodola theorem, see Herwig and Kautz (2007), the exergy loss is $T_\infty \dot S_{\rm D}$, the coefficients c_D and K must be multiplied by T_∞/T_m in order to get a measure for the loss of exergy.

If, for example, on the high temperature level in a steam power plant (T_m = 900 K) dissipation occurs in a pipe, the exergy loss with T_∞ = 300 K is only 1/3 of the dissipated mechanical energy and only this part is lost for a conversion into mechanical energy in the turbine.

The scope of this paper is to provide a comprehensive, unifying definition of loss coefficients for internal as well as external flows. This can be accomplished by relating drag and total head losses to the entropy generation rate in the flow field. Since this $\dot S_{\rm D}$ is closely related to the second law of thermodynamics, this approach is called second law analysis (SLA). A literature review with respect to this special aspect should, however, be accompanied by references about entropy in general.

2.3.1. Literature about entropy in general.

2.3.2. Literature about entropy generation.

2.3.3. Own literature about the SLA-approach.

The pivotal question in the SLA approach is how to determine the local entropy generation rate $\dot S_{\rm D}'''$ in a flow field, the integration of which leads to the (overall) entropy generation $\dot S_{\rm D}$ in equations (5) and (9). When the flow is laminar the determination of $\dot S_{\rm D}'''$ is straightforward based on the thermodynamics relation (Cartesian coordinates; units: W m⁻³ K⁻¹; see e.g. Herwig and Kautz 2007):

$$\begin{eqnarray} &&\fl\dot S_{\rm D}''' =\frac{\mu}{T} \left(2 \left[ \left(\frac{\partial u}{\partial x}\right)^2 +\left(\frac{\partial v}{\partial y}\right)^2 +\left(\frac{\partial w}{\partial z}\right)^2 \right] \right. \nonumber \\ &&+ \left.\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right)^2 \right). \end{eqnarray} \tag{ 10 }$$

In order to end up with the overall entropy generation rate $\dot S_{\rm D}$ which appears in c_D and K, see table 1, this local quantity has to be integrated appropriately. This 'appropriate' integration needs some special considerations when the flow losses of a conduit component in an otherwise fully developed internal flow should be determined, see section 3.2 below.

When the flow is turbulent $\dot S_{\rm D}'''$ according to (10) is adequate only for a direct numerical simulation (DNS)-approach with respect to the turbulence, as for the example shown in Kis and Herwig (2011). Since DNS solutions with their extraordinary computational demand cannot be used for solving technical problems, the time-averaged equations (Reynolds-averaged Navier–Stokes) are solved instead. Then also $\dot S_{\rm D}'''$ has to be time averaged, so that with $\dot S_{\rm D}'''$ as time-averaged quantity from now on

$$\begin{equation} \dot S_{\rm D}''' = \dot S_{\overline{{\rm D}}}'''+\dot S_{{{\rm D}}'}''' \end{equation} \tag{ 11 }$$

with the index $\overline {\rm D}$ for the entropy generation in the time-averaged velocity field and D' for the time-averaged contribution of the fluctuating parts. Both parts are

$$\begin{eqnarray} &&\fl\dot S_{\overline{\rm D}}''' =\frac{\mu}{T} \left(2 \left[ \left(\frac{\partial \overline{u}}{\partial x}\right)^2 +\left(\frac{\partial \overline{v}}{\partial y}\right)^2 +\left(\frac{\partial \overline{w}}{\partial z}\right)^2 \right] \right. \nonumber \\ &&+ \left.\left(\frac{\partial\overline{u}}{\partial y} + \frac{\partial \overline{v}}{\partial x}\right)^2 + \left(\frac{\partial \overline{u}}{\partial z} + \frac{\partial \overline{w}}{\partial x}\right)^2 + \left(\frac{\partial \overline{v}}{\partial z} + \frac{\partial \overline{w}}{\partial y}\right)^2 \right), \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&\fl\dot S_{{\rm D}'}''' =\frac{\mu}{T} \left(2 \left[ \overline{\left(\frac{\partial u'}{\partial x}\right)^2} +\overline{\left(\frac{\partial v'}{\partial y}\right)^2} +\overline{\left(\frac{\partial w'}{\partial z}\right)^2} \right] \right. \nonumber\\ &&+ \left.\overline{\left(\frac{\partial u'}{\partial y} + \frac{\partial v'}{\partial x}\right)^2} + \overline{\left(\frac{\partial u'}{\partial z} + \frac{\partial w'}{\partial x}\right)^2} + \overline{\left(\frac{\partial v'}{\partial z} + \frac{\partial w'}{\partial y}\right)^2} \right). \end{eqnarray} \tag{ 13 }$$

With the result for a turbulent flow field from RANS-equations, $\dot S_{\overline {\rm D}}'''$ according to equation (12) can be determined, but not $\dot S_{{\rm D}'}'''$ in equation (13). Here, a turbulence model for the fluctuating part $\dot S_{{\rm D}'}'''$ is needed.

A simple model which basically relates $\dot S_{{\rm D}'}'''$ to the turbulent dissipation rate ε and which can be justified in the limit of infinite Reynolds numbers, see Kock and Herwig (2004), reads

$$\begin{equation} \dot S_{{\rm D}'}''' =\frac{\varrho \varepsilon}{T}. \end{equation} \tag{ 14 }$$

Except for the straight pipe or channel, a conduit component has an influence on the upstream and downstream flow within certain upstream and downstream lengths. Within these lengths the influence of the component can be felt by the otherwise fully developed and undisturbed flow. In order to have a clearly defined length of considerable influence upstream and downstream, related lengths L_u and L_d are introduced. They are defined as those lengths up to which 95% of the overall upstream or downstream influence occurs.

Altogether, the situation sketched in figure 2 appears: the central part of the flow field, V_c, being the interior of the conduit component and upstream as well as downstream parts of it, V_u and V_d have to be analyzed with respect to the (additional) entropy generation due to the component. This leads to

$$\begin{equation} \underbrace{\dot S_{\rm D}\hspace{-3ex}\phantom{\int_{V_{{\rm c}}}}}_{\varphi \dot m / T_{{\rm m}}} = \underbrace{\int_{V_{{\rm u}}} (\dot S_{\rm D}''' - \dot S_{{\rm D} 0}''')\, {{\rm d}} V}_{\Delta\dot S_{{\rm D},\mathrm{u}}=\Delta\varphi_{\mathrm{u}} \dot m / T_{{\rm m}}} + \underbrace{\int_{V_{{\rm c}}} \dot S_{\rm D}'''\, {{\rm d}} V}_{\dot S_{{\rm D},\mathrm{c}}=\varphi_{\mathrm{c}} \dot m / T_{{\rm m}}} + \underbrace{\int_{V_{{\rm d}}} (\dot S_{\rm D}''' - \dot S_{{\rm D} 0}''') \,{{\rm d}} V}_{\Delta\dot S_{{\rm D},{\rm d}}=\Delta\varphi_{\mathrm{d}} \dot m / T_{{\rm m}}}. \end{equation} \tag{ 15 }$$

In equation (15) referring to figure 2 the overall entropy generation rate is shown to be composed of three parts, the rate within the conduit component and the additional rates upstream and downstream of it. Here, $\dot S_{{\rm D} 0}'''$ is the local entropy generation rate of the fully developed and undisturbed flow. Results show that always the upstream influence $\Delta \dot S_{{\rm D},\mathrm {u}}$ is very small but that the downstream influence $\Delta \dot S_{{\rm D},{\mathrm { d}}}$ can be considerable or even be the dominating part in $\dot S_{\rm D} =\Delta \dot S_{{\rm D},\mathrm {u}} + \dot S_{{\rm D},\mathrm {c}} + \Delta \dot S_{{\rm D},{\mathrm { d}}}$, see equation (9).

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It should be noted that the upstream and downstream parts V_u and V_d have lengths larger than L_u and L_d defined in figure 2. They must be chosen such that the integrals in (15) account for all $\dot S_{\rm D}''' - \dot S_{{\rm D} 0}'''$ with an acceptable truncation error.

The external flow around a body in a real fluid will always give rise to a drag force. This force has a moving point of action and thus work will be done in the flow field (in contrast to the d'Alembert paradox situation, see section 1 of this paper).

Now that the flow is one of a real fluid with non-zero viscosity, dissipation occurs which is the transfer of mechanical to internal energy. Now the work done by a steadily moving drag force can be found in a permanent increase of the internal energy of the flow field, accompanied by a corresponding entropy generation.

Within the flow field the overall entropy generation rate exactly corresponds to the drag exerted on the body, see equation (5) for c_D and table 1. Therefore $\dot S_{\rm D}$ in the flow field must be determined, just like for internal flows.

While internal flow fields are unbounded only in the upstream and downstream parts, external flow fields have no direct bounds all around. We define that part of the flow field that has to be taken into account in order to determine $\dot S_{\rm D}$ up to a set error as the near flow field. This corresponds to setting certain upstream and downstream lengths in V_u and V_d for internal flows as described in the previous sub-section.

In figure 3 K-values of a 90° bend with a circular cross section are shown for turbulent flow at various Reynolds numbers taken from Schmandt and Herwig (2011b). The K-value according to equation (9) in figure 3(a) is compared to literature values. In figure 3(b) the cross sectional entropy generation rate $\dot S_{\rm D}'$ is shown along the centerline coordinate s_c. Here the 90° bend lies between s_c/D_h = 5 and 5 + π/2. The large area of $\Delta \dot S_{{\rm D},{\mathrm { d}}}$ shows that most of the losses occur downstream of the bend. Therefore a K-value does not correspond to the losses in a conduit component but due to it.

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Laminar flow over a flat plate of finite length is a benchmark case for the higher order boundary layer theory, described in Van Dyke (1975), for example. This theory is an asymptotic theory for Re → ∞, the results of which can be applied as approximate solutions for finite values of the Reynolds number. Especially when higher order terms are taken into account, the approximation is excellent even for Reynolds numbers of the order $\mathcal {O}(1)$.

Such a higher order result for the flat plate of finite length L is

$$\begin{equation} {c_{\rm D}}=2\left[1.328 {Re}^{-1/2} + 2.67 {Re}^{-7/8} \right] + \mathcal{O}({Re}^{-1}). \end{equation} \tag{ 16 }$$

Here the second term on the rhs comes in as the influence of the trailing edge determined by the so-called triple deck theory, see for example Messiter (1983) for more details.

The drag coefficient according to equation (16) will be compared to the results we get from our SLA approach, integrating the local entropy generation in the near flow field. This is done by discretizing the two dimensional steady Navier–Stokes equations with a grid shown in figure 4. There is a grid refinement in the vicinity of the flat plate with details of the grid structure and the numerical solution procedure given in the appendix of this study.

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Like in figure 3 for the internal flow example we show the cross section integrated entropy generation rate $\dot S_{\rm D}'$ along the main stream direction in figure 5, here for the three Reynolds numbers Re = 2, 32 and 512. Comparing the three cases, one can see the increasing boundary layer character for increasing Reynolds numbers: entropy generation (and thus flow losses)occurs more and more in the vicinity of the plate (and in the wake).

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At Re = 512 there is almost no upstream influence. Since entropy generation only occurs where velocity gradients are, see equation (10), an almost undisturbed upstream flow field has almost zero $\dot S_{\rm D}'$ values. Like with internal flows, the entropy generation and thus the losses in the flow field are shifted downstream when the Reynolds number is increased.

Figure 6 shows c_D-data determined by the SLA-approach compared to the asymptotic results according to equation (16). Slightly increasing deviations for Re → 0 may be explained by the missing higher order terms $\mathcal {O}({Re} ^{-1})$ in the asymptotic result in equation (16).

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While the flat plate is a prototype from the class of slender bodies (without flow separation) the sphere is one from the class of blunt bodies (with separation, at least for high enough Reynolds numbers). Again there is an asymptotic result that can be used for comparison, this time for Re → 0. Then there is a creeping flow situation with

$$\begin{equation} {c_{\rm D}}=24/{Re} \end{equation} \tag{ 17 }$$

for the drag coefficient, see for example Van Dyke (1975).

Figure 7 shows the numerical grid for the near flow field solution of the same equations we used in the previous example (details again in the appendix), this time, however, in cylindrical coordinates.

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From the solution again the cross sectional averaged entropy generation rate $\dot S_{\rm D}'$ is shown in figure 8 for three Reynolds numbers Re = 0.1, 4 and 128. For Re = 0.1 the symmetry of the creeping flow solution for Re → 0 is well established, showing the same influence upstream and downstream.

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For increasing Reynolds numbers, convection in the downstream direction more and more shifts the entropy generation downstream since deviations from the undisturbed uniform flow (without velocity gradients, i.e. without local entropy generation) can be found predominantly downstream. At Re = 128 there is a strong asymmetry already with flow separation behind the sphere. Figure 9 compares the c_D data from our SLA-approach and the asymptotic result for Re → 0, with a clear trend of coincidence for small Reynolds numbers.

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In order to show the wide applicability of the SLA-approach and its physical significance, the third example for external flows is the flow around a rising air bubble in water and the drag involved. Again, a drag coefficient can be determined by integrating the entropy generation in the near flow field around the upward moving bubble.

When a single bubble has a diameter d < 1.5 mm the high surface tension leads to an almost perfect spherical shape of the bubble and the path it takes on its way upwards is straight, see Peters and Gaertner (2011) for details. Such a case can be modeled by a fixed bubble with a homogeneous oncoming flow in the direction of the gravity vector $\vec g$. At the bubble surface no shear stress occurs since the gas motion inside the bubble has a negligible momentum due to the low density compared to that of the surrounding water. Then, however, the bubble surface is totally mobile which can be accounted for by a total slip velocity boundary condition at the surface of the spherical bubble.

Due to the change in boundary conditions from non-slip to slip the velocity gradients near the sphere and bubble surfaces are fundamentally different, and—as a consequence—also the local entropy generation rates.

This can be demonstrated for the cross section averaged entropy generation rates $\dot S_{\rm D}'$ in figure 10 compared to the corresponding distributions for the sphere in figure 8. Without no-slip condition there is no boundary layer at the bubble surface and even for higher Reynolds numbers no flow separation.

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For spherical bubbles there are asymptotic solutions for Re → 0, associated with the names Hadamard and Rybczinsky, see Clift et al (1978), and for Re → ∞, see Levich (1962). They are

$$\begin{equation} {c_{\rm D}}= 16/{Re} \quad {\rm for}\; {Re}\to0,\end{equation} \tag{ 18 }$$

$$\begin{equation} {c_{\rm D}}= 48/{Re} \quad {\rm for}\; {Re}\to\infty. \end{equation} \tag{ 19 }$$

In figure 11 both asymptotes are shown together with our SLA-results (numerical details are given in the appendix). There is a smooth transition between the asymptotic results for small and large Reynolds numbers showing that the integrated entropy generation corresponds to the losses in the flow field for the whole Reynolds number range of laminar flow.

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A comparison with experimental data in oil, see Peters and Gaertner (2011), as a further validation of the SLA-approach is given in figure 12, here with a close coincidence of experimental and theoretical data.

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For a turbulent internal flow through a bend, developed flow is assumed to prevail at the beginning of the numerical domain. Therefore, profiles for the velocity and the turbulent quantities k and ω for a low Re–k–ω–SST model have to be provided. They are taken from previous simulations of channel flow with periodic boundary conditions for this case, see Schmandt and Herwig (2011b) for details. At the walls, the no-slip boundary condition for the velocity, i.e. $\vec {u}=[0,0,0]$ is applied. Appropriate values for the turbulent quantities are automatically provided by the turbulence model. The level of the so-called kinematic pressure $\hat p = p/\varrho$ is set via a fixed value $\hat p=0$ adequate for incompressible flow. The pressure at the inlet then is a result of the calculation.

Since laminar flow is assumed, the flow situation is given by a far field velocity corresponding to the Reynolds number and the characteristic length scale, only. The velocity $\vec u=[u_\infty ,0,0]$ is given at an upstream cross-section perpendicular to the direction of the main flow. At an opposing downstream cross section the normal gradient of the velocity is set to zero. For the remaining boundary parallel to the direction of the main flow, a slip boundary condition is applied. This boundary condition enforces a parallel flow in the far field. This, however, needs special attention since all components generally lead to a lateral displacement of the massflow. This problem can be circumvented using a numerical domain with a large enough expansion in the direction orthogonal to the direction of the main flow, so that the influence of the body on the far field is small.

To minimize the grid size, two-dimensional flow and symmetry are assumed for the flat plate while the flow around the sphere and the bubble is assumed to be axi-symmetric. In the latter case, only a wedge with an angle of 5° has to be covered by the grid resulting in a size reduction by a factor 72 compared to a full three-dimensional grid.

The grids used and displayed in the previously described examples from section 4 are generated using the blockMesh utilities from the respective OpenFOAM distributions. This tool is able to generate grids with cells of hexagonal topology. The cells are described via several structured blocks with a prescribed number of divisions in the three spatial directions of each block. Yet, the resulting grid is stored in an unstructured manner.

For various flow situations it is possible to define grids with flat cell layers perpendicular to the direction of the main flow. Among these are the internal flows through smooth pipe bends. With these layers an integration of $\dot S_{\rm D}'''$ at a certain position $\tilde x/D_{\mathrm { h}}$ along the path of the main flow can be computed easily. The integral then becomes the scalar product of the $\dot S_{\rm D}'''$-field and the scalar field of the single cell volumes, each belonging to cells with cell center coordinates $x<\tilde x$. Therefore, $\tilde x$ has to be aligned according to the cell layers. This leads to values $[\dot S_{\rm D}]_{x=0}^{x=\tilde x} = \dot S_{\rm D} (\tilde x)$. A subsequent derivation with respect to x then gives $\dot S_{\rm D}'(x)$, see figure 3.

For the flow around the spheres and bubbles, however, it is not possible to generate grids of acceptable quality with cell layers perpendicular to the x-direction. In these cases, some cells are sliced by the clip planes at discrete positions $\tilde x$, now at arbitrary positions not aligned with cell layers. The contribution of the split cells to $\dot S_{\rm D}(\tilde x)$ has to be regarded via reduced volumes in addition to the complete cells upstream of $\tilde x$. Values of $\dot S_{\rm D}'''$ are always assumed to be constant within each cell. The upstream parts of the divided cell volumes are computed using the convhulln function distributed with Matlab, which is used for the fully automated post-processing. As function input to the convhulln function the grid points of the cut cells located upstream of the clip plane are provided together with the intersecting points of the cell edges with the clip plane. Then the partial volumes appear as an outcome of the convex hull described by these points. Volumes of complete cells, however, are accessed by a customized OpenFOAM utility at much lower computational expense. The whole procedure is enabled by OpenFOAM which stores the grid data as easily accessible compressed ASCII-files. Values, which are not stored in the grid files, like the cell center coordinates, can be computed from the grid data via OpenFOAM utilities and are automatically read by the Matlab post-processing script.