A Finite Volume Method for Pressure Extraction on...

18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 – 7, 2016

A Finite Volume Method for Pressure Extraction on Unstructured Flow Data

N. J. Neeteson1, *, D. E. Rival1 1: Mechanical and Materials Engineering, Queen's University, Kingston, Canada

* Correspondent author: [email protected]

Keywords: Pressure extraction, Lagrangian flow data, Finite volume method

ABSTRACT

A novel, finite volume method-based, Lagrangian pressure-extraction technique (Lagrangian FVM) is proposed and comparatively evaluated using an analytical test case. First, the derivation of the finite-volume formulation of the Poisson equation for pressure is presented, along with a detailed explanation of how the method is implemented. Next, the analytical test case, the Taylor-Green vortex field, is defined. Over a range of spatial particle densities, and using mixed Neumann and Dirichlet boundary conditions, the performance of the Lagrangian FVM is compared to a Lagrangian pressure-extraction technique based on finite-differences (Lagrangian FDM), as well as a basic second-order Eulerian pressure-extraction technique. Over the range of spatial particle densities surveyed, the Lagrangian FVM is found to estimate the pressure field with one half of the error as compared to the Eulerian technique, and an order of magnitude less error as compared to the Lagrangian FDM. Further, Lagrangian FVM produces the most consistent error fields, both spatially and between trials. Finally, a path forward for further investigation of the technique is described, including further comparative tests with high-fidelity experimental data, and a more thorough investigation into the propagation of errors in the underlying flow data.

1. Introduction 1.1 Background The instantaneous pressure field is of practical interest in many sub-domains of fluid dynamics, including turbulence studies (Ghaemi et al. 2012), investigations of cavitation dynamics (Liu and Katz 2006), acoustics (Haigermoser 2009), and it is of critical importance in the estimation of dynamic loads on submersed bodies (Fujisawa et al. 2005, Murai et al. 2007). While methods exist for directly measuring the pressure at points in a fluid using physical probes, these methods require a physical presence in the fluid, and their temporal resolution is limited (Chue 1975, van Oudheusden 2013). The advantages of indirect pressure-measurement techniques are twofold: they are non-intrusive, and the spatial and temporal resolution of the indirect pressure measurement corresponds to that of the direct optical measurements. Numerous novel and unique methods of non-intrusive pressure-field measurement have been developed and tested, such as


the use of pressure-sensitive paints (McLachlan and Ball 1995), however, the focus of this work is on the development of methods for extracting the pressure field from experimentally-measured velocity fields. Since the seminal works of (Schwabe 1935), (Gurka et al. 1999), (Baur and Kongeter 1999), and many others, there has been a great deal investigation into increasingly sophisticated velocimetry-based pressure-measurement techniques (van Oudheusden 2013). Historically, Eulerian flow-measurement techniques have dominated Lagrangian flow-measurement techniques in terms of spatial resolution (Raffel et al. 2007, Lüthi et al. 2005), and so it is not surprising that, to this point, the vast majority of investigations into velocimetry-based pressure measurement have been concerned with Eulerian flow measurements in an Eulerian frame (van Oudheusden 2013). Very recently, the rate of advancement of Lagrangian measurement techniques has accelerated rapidly, to the point that it is now possible to collect Lagrangian flow data with a spatial resolution and accuracy that is equal to, if not superior to, that of Eulerian flow data (Wieneke 2013, Schanz et al. 2016). Considering that Lagrangian flow data allows direct calculation of the material derivative of velocity, a vector quantity that is critical in pressure extraction, and the established advantages of Lagrangian measurement techniques over Eulerian measurement techniques in near-wall regions (Kähler et al. 2012), there is clear motivation for the investigation of pressure-extraction techniques that capitalize on the inherent advantages of Lagrangian flow data. There have been several recent studies incorporating true and pseudo Lagrangian velocity measurements, from particle tracking velocimetry and/or trajectory reconstruction, into Eulerian pressure-extraction techniques (Violato et al. 2011, Novara and Scarano 2013, Jeon et al. 2014). To this point, however, there has been a distinct lack investigation into pressure-field extraction in the Lagrangian frame. To fill in this apparent dearth in the literature, the authors of the present work have recently proposed and demonstrated a novel Lagrangian pressure-extraction technique. In this Lagrangian pressure-extraction technique, a network is constructed on the Lagrangian flow data and the governing differential equations are approximated on this network using the finite difference method, with shape functions based on the geometry of the network, circumventing the step of interpolating the Lagrangian data to an Eulerian structured grid (Neeteson and Rival 2015). As this technique used finite differences to approximate derivatives, it is hereafter referred to as the Lagrangian FDM for pressure extraction. Upon further investigation, the technique was found to perform well with Dirichlet boundary conditions, but suffered from significant increases in


pressure-extraction error when Neumann boundary conditions were applied on boundaries with strong pressure gradients (Neeteson et al. 2016). In the present study, a novel, finite volume-based technique for Lagrangian pressure extraction is proposed: the same Lagrangian network is used, but instead the governing differential equations are converted to an integral form and approximated on the network using the finite volume method. The purpose of this study is first to describe the approximation and discretization of the Lagrangian finite volume method (Lagrangian FVM), and second to compare the performance of the Lagrangian FVM to both the previous Lagrangian pressure extraction technique (Lagrangian FDM) and a typical Eulerian Poisson solver approximated using second-order finite differences. 1.2 The Lagrangian Network In order to extract the pressure field entirely in the Lagrangian domain, it is first necessary to construct a network on which vector calculus operations can be performed. In the Eulerian domain, the construction of a network is trivial: points are simply connected to their immediate neighbours in each of the cardinal directions with which the structured grid is aligned. For a uniform structured grid, all shape functions are equal and all control volumes are cubes with the same side length, making it trivial to approximate differential equations using finite-difference

methods or integral equations using finite-volume methods. Conversely, an unstructured

(a) (b) (c)

Fig. 1 The basic unit of the Lagrangian network, shown in 2D for simplicity. (a) A Delaunay triangulation connects

the center point to its neighbours, and a Voronoi tessellation creates cells around the points. (b) The control volume

corresponding to the center point is shown shaded in blue, and the control surface highlighted in green. (c) For a

given facet of the control surface, the point-to-neighbour (𝑖-to-𝑗) distance is ℎ$% , the corresponding control surface

area is 𝑠$% , and the unit vector pointing from 𝑖 to 𝑗 is 𝑛($% .


Lagrangian network will produce non-unity shape functions and multi-faceted control volumes, with each network being unique to the set of Lagrangian spatial coordinates on which it is constructed. In the present study, this Lagrangian network is constructed using the Delaunay triangulation and Voronoi tessellation, geometric operations which connect a network of points, and construct a patchwork of finite cells corresponding to those points, respectively (Aurenhammer 1991). A detailed description of the construction of this network, and the equations for approximating the various vector calculus operators that are required, are given in the original study that proposed and demonstrated Voronoi network-based pressure extraction (Neeteson and Rival 2015). It is important, however, to emphasize a few important qualities of a Delaunay/Voronoi-based network: First, the Delaunay triangulation seeks to connect the points in such a way that the number of `skinny' triangles is minimized, that is to say that the skewness of the network is minimized. Second, the Voronoi tessellation constructs cells around each point, hereafter referred to as control volumes, such that each facet of the control surface corresponds to a single connected neighbour and is oriented normal to the vector pointing from the central point to the neighbour. Additionally, each facets of each control volume is equidistant to the two corresponding center nodes, ensuring that approximations made on the control surface using either center differences or linear interpolation are second-order. A simplified two-dimensional diagram of the Lagrangian network is shown in figure 1. Each connection in the Lagrangian network has three important properties associated with it: 𝑛$%, the unit vector pointing from 𝑖 to 𝑗, ℎ$%, the distance from 𝑖 to 𝑗, and 𝑠$%, the area of the facet of the control surface corresponding to 𝑖 and 𝑗. Based on the definition of how the network is constructed, it can additionally be observed that 𝑛$% = −𝑛%$, ℎ$% = ℎ%$, and 𝑠$% = 𝑠%$. These identities are not used in the derivation or execution of vector calculus operators on the network, but they are useful in speeding up the construction of the network. 1.3 The Lagrangian Finite Volume Method for Pressure Extraction While the previous treatments on Voronoi-network-based pressure extraction utilized finite difference methods to approximate the incompressible vector differential Navier-Stokes equation to the Lagrangian network (Neeteson and Rival 2015, Neeteson et al. 2016), the present study instead tackles the problem using the finite volume method. The derivation of the governing


system of discretized equations begins with the differential form of the incompressible Navier-Stokes equation: 𝜌

𝐷𝑢𝐷𝑡 = −∇𝑝 + 𝜇∇

4𝑢, (1)

where 𝜌 is the fluid density, 𝑢 is the velocity vector field, ∇ is the vector differential operator, 𝑝 is the pressure field, and 𝜇 is the fluid viscosity. Applying the divergence operator to obtain Poisson's equation for pressure (Gurka et al. 1999) produces the Lagrangian formulation of Poisson's equation for pressure (Neeteson and Rival 2015): ∇4𝑝 = −𝜌∇ ⋅

𝐷𝑢𝐷𝑡 .

(2)

If this expression holds for an infinitesimal point in the field, it must also hold for a finite region, therefore it can be integrated over a control volume with no loss of generality: ∇4𝑝𝑑𝑉

:= −𝜌 ∇ ⋅

𝐷𝑢𝐷𝑡 𝑑𝑉:

, (3)

where Ω refers to the control volume. Divergence theorem can then be applied to reduce the order of the spatial derivatives: (∇𝑝 ⋅ 𝑛)𝑑𝐴

?:= −𝜌

𝐷𝑢𝐷𝑡 ⋅ 𝑛 𝑑𝐴

?:, (4)

where 𝜕Ω refers to the boundary of the control volume, or the control surface. This equation can now be discretized using standard second-order finite volume methods - specifically by representing each surface integral as a summation over the facets of a control volume, using linear interpolation to obtain the value of the material derivative at control surfaces, and using a central-difference scheme to approximate the pressure gradient on control surfaces. Since every facet of a given control volume is equidistant to the two corresponding nodes, all approximations on the control surfaces are second-order. The final discretized form of the system is: 𝑠$%

𝑝% − 𝑝$ℎ$%

A

%= 𝜌

𝑠$%2

𝐷𝑢𝐷𝑡

$+

𝐷𝑢𝐷𝑡

%⋅ 𝑛$%

A

%. (5)

Finally, the treatment of boundary conditions must be defined. Dirichlet boundary conditions will be treated explicitly, meaning that Dirichlet boundary conditions will be enforced on control volumes that remain active. These control volumes will be referred to as Dirichlet control volumes. Conversely, Neumann boundary conditions will be treated implicitly, meaning that Neumann boundary conditions will be enforced on active control volumes adjacent to Neumann boundaries, and control volumes on Neumann boundaries will be deactivated. These control volumes will be referred to as Neumann control volumes. A new network parameter 𝛼$% is introduced, and defined as such: 𝛼$% =

1,0,𝑖𝑓𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑗𝑜𝑓𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑣𝑜𝑙𝑢𝑚𝑒𝑖𝑖𝑠𝑎𝑁𝑒𝑢𝑚𝑎𝑛𝑛𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑣𝑜𝑙𝑢𝑚𝑒.𝑖𝑓𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑗𝑜𝑓𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑣𝑜𝑙𝑢𝑚𝑒𝑖𝑖𝑠𝑛𝑜𝑡𝑎𝑁𝑒𝑢𝑚𝑎𝑛𝑛𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑣𝑜𝑙𝑢𝑚𝑒.

(6)


The left hand side of equation 5 is split apart into two terms, multiplied by 𝛼$% and (1 − 𝛼$%), respectively. The former term remains on the left hand side and in its present form, while the latter term is moved to the right hand side, as this is the term which will be used to implicitly enforce Neumann boundary conditions. To do this, the approximate pressure gradient, calculated using equation 1, is integrated over the Neumann control surfaces. After some algebra, the final discretized system is: 𝛼$%

𝑠$%ℎ$%

A

%𝑝$ − 𝛼$%

𝑠$%ℎ$%

𝑝%A

%

= 𝜌𝑠$%2

𝐷𝑢𝐷𝑡

$+

𝐷𝑢𝐷𝑡

%⋅ 𝑛$%

A

%+ 1 − 𝛼$%

𝑠$%2 ∇𝑝

$+ ∇𝑝

%⋅ 𝑛$%

A

%

(7)

where ∇𝑝$, ∇𝑝

% refer to the pressure gradient calculated using equation 1 at point 𝑖 and 𝑗,

respectively. The left hand side consists only of constant coefficients multiplied by the pressure field, and the right hand side consists entirely of constants that can be calculated from the Lagrangian flow data. Therefore, the system is now in the form 𝐴$%𝑝% = 𝑏$, and the pressure field can be extracted using one of the many indirect or direct methods for solving linear systems of equations. In the present study, the generalized minimal residual method (gmres) is used to solve systems of linear equations (Saad and Schultz 1986). 2. Analytical Case 2.1 The Taylor-Green Vortex Field An analytical solution to the Navier-Stokes equations can be used to quantify the error in the pressure field extracted from a particular velocity and acceleration field. The analytical solution used in this study is the instantaneous Taylor-Green vortex field (Taylor and Green 1937), which is shown in figure 2, and which can be described by the following closed-form expressions: 𝑢R = 𝑉S sin

xL cos

yL cos

zL , (8)

𝑢] = 𝑉S cosxL sin

yL cos

zL , (9)

𝑢^ = 0, (10)

𝑝 =𝜌𝑉S4

16 cos2𝑥𝐿 + cos

2𝑦𝐿 cos

2𝑧𝐿 + 2 , (11)


where 𝑢$ is the 𝑖de component of velocity, 𝑉S is the maximum velocity, 𝐿 is a characteristic length scale, and 𝜌 is the fluid density. A closed-form expression for the three-dimensional components of the material derivative of velocity can be obtained by substituting these expressions for pressure and velocity into equation 1 and isolating for the vector material derivative of velocity. The spatially-periodic nature of the Taylor-Green vortex field provides two additional advantages in the quantitative evaluation of pressure-extraction techniques: First, as can be observed in figure 2, a domain can be chosen such that multiple minima and maxima of pressure are present in the domain, which allows a given pressure-extraction technique to be evaluated with respect to how well it captures multiple pressure minima and maxima in close proximity. Second, the boundaries of the synthetic measurement domain can be chosen to intersect with vortices, maximizing the value of the pressure gradient on the domain boundary. This allows the implementation of the Neumann boundary conditions to be tested in the most unfavorable conditions possible. Since the Lagrangian pressure-extraction technique based on a finite-difference method was shown to perform poorly when strong pressure gradients were present on Neumann boundaries (Neeteson et al. 2016), it is critical that the new Lagrangian pressure extraction technique be evaluated in these same conditions. Furthermore, having a single boundary of the domain, or less, correspond to a Dirichlet boundary condition, with the rest of the

Fig. 2 The analytical pressure field of the Taylor-Green vortex field at 𝑡 = 0, shown as a 2D slice at 𝑧 = 0. The flow

field and pressure field are both spatially periodic and the pattern repeats infinitely in all directions. Pressure fields

extracted from synthetic Lagrangian flow data will be compared to this analytical pressure field in order to

quantify the pressure-extraction error.


domain boundary corresponding to a Neumann boundary condition, is the most experimentally-relevant boundary condition configuration. 2.2 Pressure-Extraction Methods First, synthetic Lagrangian flow data was generated for a range of spatial particle densities. Analytical tests were conducted in a cubic region with side length 3𝜋 centered at the origin, a 2D slice of this domain is shown in figure 2. This domain was selected to ensure that several vortical structures were present within the synthetic measurement domain, and that there were strong pressure gradients present on all domain boundaries. The flow data was generated by assigning particle positions using uniform random data and calculating the velocity and material derivative corresponding to each particle using the closed-form expressions given in the preceding subsection. At each spatial particle density, three separate Lagrangian datasets were generated and used, in order to smooth out any random errors introduced by local inhomogeneities in the distribution of the Lagrangian particles. For visualization purposes a single trial was arbitrarily picked, but for assessing the accuracy of the techniques versus the spatial particle density, the mean field errors in each trial were averaged together. This gives an idea of the average performance that can be expected from a given technique. In order to perform a preliminary comparative analysis, three pressure-extraction techniques were used to extract the pressure field from the generated synthetic flow data: a standard Eulerian-frame, finite-differences-based Poisson solver (van Oudheusden 2013), the previously-proposed Lagrangian FDM technique (Neeteson and Rival 2015, Neeteson et al. 2016), and the newly-proposed Lagrangian FVM pressure-extraction technique. These three techniques are briefly described in the following sub-sections. In all three methods, the boundary conditions used were the same. Dirichlet boundary conditions were applied on the top (𝑧h) of the domain, and Neumann boundary conditions were applied on the five remaining faces of the cubic domain. Boundary conditions were enforced implicitly in all three methods, meaning that the Eulerian grid points, Lagrangian grid points, and Lagrangian control volumes on the boundary of the domain are ignored in the solution stage. Rather than calculating the pressure on the boundary, the prescribed pressure or pressure gradient from the boundary is inserted into the discretized governing equation for points (or control volumes) adjacent to boundaries. In this analysis boundaries were ignored entirely, but in a case where the


pressure field on the boundary is desired, one could extrapolate the pressure field from the interior to the boundary using the approximate pressure-gradient field. 2.2.1 Eulerian Pressure Extraction The first step in the Eulerian pressure-extraction process is to interpolate the Lagrangian flow data, which includes both the velocity field and the Lagrangian acceleration field, to an Eulerian grid. The Eulerian grid is constructed such that the grid density was approximately equivalent to the Lagrangian particle density - mathematically this can be described by calculating the side length of an Eulerian voxel as: 𝑝 =

𝑉𝑁i

j/l

, (12)

where 𝑉 is the volume of the domain and 𝑁i is the number of Lagrangian particles. Correspondingly, the number of Eulerian grid points is then: 𝑁m =

𝑉j/l

𝑝

l

, (13)

Simple natural-neighbour interpolation is used to convert the Lagrangian data to Eulerian data (Sibson 1981). Following the interpolation, the source field for the Poisson equation for pressure and the approximate pressure-gradient field are calculated on the Eulerian grid using second-order finite difference approximations, replacing the Eulerian material derivative of velocity with the interpolated Lagrangian particle acceleration field wherever necessary. Finally, the pressure field is extracted using a simple iterative Poisson solver (van Oudheusden 2013). 2.2.2 Lagrangian FDM Pressure Extraction The Lagrangian FDM pressure-extraction procedure is described in detail in the two previous studies in which it was developed and tested (Neeteson and Rival 2015, Neeteson et al. 2016). First, the Delaunay triangulation and Voronoi tessellation are applied to the Lagrangian particle field in order to both construct the network and compute the required network parameters ℎ$%, 𝑠$%, 𝑛$%. Next, the source field for the Poisson equation for pressure and the approximate pressure-gradient field are calculated using Lagrangian finite-difference approximations on the network, by taking the divergence of the Lagrangian acceleration field, and using equation 1, respectively. Finally, a Lagrangian iterative Poisson solver, derived from a Lagrangian approximation for the Laplacian operator, is used to extract the pressure field. No additional points are added to the domain to


enforce boundary conditions. Instead, any points located within 0.15𝜋 of the domain boundary are designated as boundary points. 2.2.3 Lagrangian FVM Pressure Extraction As in the Lagrangian FDM method, the first step is to apply the Delaunay triangulation and Voronoi tessellation to the Lagrangian particle field in order to construct the network and compute the required network parameters. Next, the Lagrangian pressure-gradient field is calculated using Lagrangian finite-difference approximations to approximate equation 1. Finally, the pressure field is solved by using equation 7 to set up a system of linear equations, which are solved using the gmres algorithm. Just as in the Lagrangian FDM method, any points located within 0.15𝜋 of the domain boundaries are designated as boundary control volumes, which are ignored in the solution process and all subsequent analysis. 2.3 Pressure-Extraction Results Both the spatial particle densities and pressure fields were normalized before analysis. The pressure was normalized by using the familiar definition of the pressure coefficient: 𝐶p =

𝑝 − 𝑝S12 𝜌𝑉S4

, (14)

where 𝐶p is the pressure coefficient, 𝑝 is the static pressure, 𝑝S is the free-stream pressure, taken to be 0 for this analysis, 𝜌 is the fluid density, and 𝑉S is the characteristic velocity, typically the free-stream velocity, but in this case taken to be the maximum velocity in the Taylor-Green vortex field. The spatial particle density is normalized as such: 𝑁∗ =

𝑁𝐿l

Ω , (15)

where 𝑁∗ is the normalized spatial particle density, 𝑁 is the number of Lagrangian particles in the field, 𝐿 is a characteristic length scale of the flow structure to be resolved, or of the flow in general, and Ω is the volume of the domain. The spatial particle density can be thought of as the number of Lagrangian particles per characteristic volume (Neeteson et al. 2016).


Figure 3 shows a log-log plot of the mean normalized pressure-extraction error across the domain, 𝛿𝐶p, as a function of the normalized spatial particle density for the three pressure extraction techniques examined in the analytical study. It can be observed that the mean field error in both the Eulerian and Lagrangian FVM pressure-extraction techniques have an approximate power-law dependency on the normalized spatial particle density. Additionally, over the range of spatial particle densities surveyed, the Lagrangian FVM appears to produce estimates of the pressure field with approximately half of the error as compared to estimates of the pressure field produced by a basic interpolation and iterative Eulerian Poisson solver. The random scatter in the mean field error estimates for the Lagrangian FDM pressure-extraction technique is such that it is difficult to characterize its relationship to the normalized spatial particle density. It is possible that refining the particle density increment and/or performing more trials at each particle density could refine this data series and allow the trend to be identified, however, since the principle goal of this study is to evaluate the performance of the Lagrangian FVM,

Fig. 3 Log-log plot of the mean normalized pressure-extraction error 𝛿𝐶psssss, versus the number of points per

normalized volume 𝑁∗, where the volume is normalized by the characteristic length of a vortex in the field. The

techniques shown are the Eulerian, Lagrangian FDM, and Lagrangian FVM pressure-extraction techniques. Each

data point represents a mean error from three trials, and trials were performed over a range of 37 < 𝑁∗ < 593,

increasing geometrically by a factor of 2j/4.


performing additional studies to better characterize the performance of the Lagrangian FDM was deemed unnecessary. There are three important results to take away from the comparison of the mean field error in the estimates produced by the Lagrangian FDM and Lagrangian FVM: First, the Lagrangian FVM produces estimates of the pressure field with an order of magnitude less error than the Lagrangian FDM. Second, the Lagrangian FVM is much less sensitive to variations in the spatial homogeneity of the particle field and correspondingly will produce much more reliable estimates of the pressure field. In other words, the error in the estimated field is more predictable and less dependent on the

Fig. 4 Sample pressure fields for 𝑁∗ ≈ 593, shown as a slice of the data at 𝑧 = 0 for the Eulerian data and −0.5 <

𝑧/𝐿 < 0.5 for Lagrangian data. Sub-figures show the (a) analytical reference pressure field, (b) pressure field

extracted using the Eulerian technique, (c) pressure field extracted using Lagrangian FDM, and (d) pressure field

extracted using Lagrangian FVM.


structure of the network. Third, the Lagrangian FVM does not display the issues with the Neumann boundary condition that were observed in the previous study (Neeteson et al. 2016) and which can be seen in figure 3, where the error in the Lagrangian FDM ceases decreasing with normalized spatial particle density beyond 𝑁∗ ≈ 100. These issues will be examined in slightly more detail below. Examining sample pressure and error fields can provide insight into the performance of the various pressure-extraction techniques, and figures 4 and 5 show true and estimated pressure and error fields, respectively, sampled from a single trial at 𝑁∗ ≈ 593. First examining figure 4, it can be observed that all three techniques have satisfactorily captured the overall structure of the true underlying pressure field. In each of the extracted pressure fields, all nine pressure maxima and all four pressure minima within the interior of the measurement domain are qualitatively apparent. While it may appear that the Lagrangian techniques have failed to capture the pressure minima on the boundaries compared to the Eulerian technique, this effect is actually due to the fact that the Eulerian structured domain has a thinner layer of boundary points than what was used in the Lagrangian domain (only interior points are displayed in the figures).

Fig. 5 Sample error fields for 𝑁∗ ≈ 593, shown as a slice of the data at 𝑧 = 0 for the Eulerian data and −0.5 < 𝑧/𝐿 <

0.5 for Lagrangian data. Sub-figures show the (a) error in pressure field extracted using the Eulerian technique, (b)

error in pressure field extracted using Lagrangian FDM, and (c) error in pressure field extracted using Lagrangian

FVM.


The pressure field estimated using the Lagrangian FDM can clearly be seen to be a biased estimate of the true pressure field, with the magnitude of the pressure minima being underestimated and the magnitude of the pressure maxima being overestimated. These errors are due to the propagation of errors originating at the Neumann boundaries, as observed in previous studies (Neeteson et al. 2016). For both the Eulerian procedure and Lagrangian FVM, the differences between the true and estimated pressure fields are slight enough that comparing the estimated pressure fields does not provide insight into how they differ. The normalized absolute error fields, figure 5, must be examined instead. First, the observed absolute error field produced by the Lagrangian FDM confirms that the high errors throughout the field are generated at Neumann boundaries and then propagated throughout the rest of the field. The error can be observed to be largest on the Neumann boundaries, particularly in regions where the pressure gradient is largest on the boundary. These errors diffuse out through the domain due to the elliptic nature of the Poisson equation. The error field produced by the Eulerian technique does not display a sensitivity to the boundary conditions, but rather the error peaks at the locations of the pressure maxima and minima. The Eulerian technique shows errors on the order of 𝛿𝐶p ≈ 0.1, or 10% of the pressure maxima. In the Taylor-Green vortex field, regions where the pressure is highest are coincident with regions where the velocity and acceleration fields are changing rapidly, so it is likely that these pressure-extraction errors are caused by the propagation of interpolation errors. The semi-random nature of the magnitude and form of the errors in the Eulerian pressure field further support the possibility that interpolation errors are the principle contributor to the error in the estimated pressure field. Finally, examining the error in the pressure field produced by the Lagrangian FVM, neither the pressure maxima errors of the Eulerian technique, nor the boundary-related errors of the Lagrangian FDM can be observed. In fact, it is difficult to discern any significant pattern in the distribution of error. Instead, the error appears to be distributed somewhat randomly in the domain, with small pockets of slightly-larger error that do not appear to correspond to physical flow structures. It can further be observed that the error in the pressure field extracted using the Lagrangian FVM is much more spatially consistent than the Eulerian and Lagrangian FDM pressure extraction techniques.


3 Conclusions In this study, the previously-proposed Lagrangian framework for pressure extraction on unstructured flow data (Neeteson and Rival 2015) has been updated by replacing the previous finite-difference approximations with newly-derived Lagrangian finite-volume approximations. In the absence of experimental data, an analytical test case, the Taylor-Green vortex field, is used to evaluate the performance of the Lagrangian FVM for pressure extraction relative to the baseline Eulerian Poisson technique for pressure extraction and the previous Lagrangian FDM for pressure extraction. An analysis of the relationship between the mean field error and the spatial particle density determined that the Lagrangian FVM produces pressure fields with approximately half of the mean error as the baseline Eulerian technique, and an order of magnitude less error than the Lagrangian FDM. Furthermore, the Lagrangian FVM is found to be far more consistent in its estimation of the pressure field than the Lagrangian FDM, a key quality that will allow the the error in a given extracted pressure field to be more reliably predicted in the absence of a reference pressure field. Comparing sample spatial error fields for the three cases demonstrated that the Lagrangian FVM has several advantages over the baseline Eulerian technique and Lagrangian FDM. As predicted, avoiding the step of interpolation eliminates sources of error which are detrimental to the estimation of the pressure field in the Eulerian technique. In particular, the Lagrangian FVM is better able to capture pressure maxima than the baseline Eulerian technique, capturing the pressure maxima with nearly an order of magnitude less error. Further, switching from a finite-difference to finite-volume method of discretization completely eliminates the sensitivity to pressure gradients on the boundary observed in previous studies, and because of this the Lagrangian FVM dominates the Lagrangian FDM when Neumann boundary conditions are applied. Overcoming the sensitivity to Neumann boundary conditions results in a substantial improvement to the feasibility of the technique, as Neumann boundary conditions are all but unavoidable when applying pressure-extraction techniques to real experimental data. Based on the results of this preliminary study on the Lagrangian FVM for pressure extraction, there are two clear paths forward: First, the new technique must be quantitatively evaluated using high-fidelity experimental data, preferably in comparison to a state-of-the-art Eulerian technique with the same data. Comparative


tests using analytical data and baseline Eulerian techniques can provide useful insight on the performance and validity of a novel pressure-extraction technique, but without a rigorous comparative study with real experimental data, it is impossible to comment substantively on the feasibility of the technique, or whether its advantages are sufficient to advocate that it be used in place of the commonly-used Eulerian techniques for pressure extraction. Second, it is critical that a framework or procedure be established for the estimation of the spatial pressure-extraction error on a specific Lagrangian network, in an analogous manner as was recently proposed for PIV-based pressure extraction (Azijli et al. 2016). As observed in the analytical results section, it is not clear what exactly is driving the distribution of the error in the pressure fields generated by the Lagrangian FVM for pressure extraction. It is possible that local errors are proportional to the local inhomogeneity of the particle field, but further investigation is required. Therefore, it is of critical importance that the performance of the Lagrangian FVM, particularly its response to perturbations in the velocity and acceleration fields, is analyzed to determine a method for estimating the approximate error in an estimated pressure field when no reference data is available, as is nearly always the case when applying pressure-extraction techniques to experimental data. 4 References

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