Deriving forces from 2D velocity field measurements

Eur. Phys. J. Special Topics 220, 91–100 (2013)© EDP Sciences, Springer-Verlag 2013DOI: 10.1140/epjst/e2013-01799-9

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Deriving forces from 2D velocity fieldmeasurements

Thomas Albrecht1,a, Vanessa del Campo2, Tom Weier1, Hans Metzkes3,and Jorg Stiller3

1 Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, PO Box 51 01 19,01314 Dresden, Germany

2 ETSEIAT, UPC, Polytechnic University of Catalunya, 08222 Terrassa, Spain3 Institute of Fluid Mechanics, Technische Universitat Dresden, 01062 Dresden, Germany

Received 17 December 2012 / Received in final form 5 February 2013Published online 26 March 2013

Abstract. We discuss how to derive a force or a force density from ameasured velocity field. The first part focuses on the integral forcea fluid exerts on a body, e.g. lift and drag on an airfoil. Obtain-ing the correct pressure is crucial; however, it cannot be measuredwithin the flow non-intrusively. Using numerical and experimental testcases, we compare the accuracy achievable with three methods: pressurereconstruction from velocity fields via (1) the differential momentumequation, or (2) the Poisson equation, furthermore, (3) Noca’s momen-tum equation [Noca, JFS 13(5), 1999], which does not require pressureexplicitly. The latter gives the best results for the lift, whereas the firstor second approach should be used for the drag. The second part dealswith obtaining the distribution of a body force density generated by anactuator. Using a stream function ansatz, we obtain a Laplace equationthat allows us to compute the solenoidal part of the force distribution;however, the irrotational part is lost. Furthermore, the wall pressuremust be known. We validate this approach using numerical data froma wall jet flow in a rectangular box, driven by a fictitious, solenoidalbody force. Reconstructing the force distribution yields an error of lessthan 10−2 for most of the domain.

1 Introduction

Standard Particle Image Velocimetry (PIV) can deliver accurate, time-resolved, two-dimensional, two-component velocity fields with comparably little effort and, mostimportantly, without disrupting the flow. Extensions such as Stereo-PIV, scanningtechniques, or light field imaging can fill in gaps to create full three-dimensional,three-component (3D-3C) measurements [1]. With these extensive data available, newpossibilities for advanced processing arise. In addition to features such as vortices,separation regions, and stagnation points, which can be more or less directly visual-ized, the flow field also includes some less obvious information: integral forces exerted

a e-mail: [email protected]

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92 The European Physical Journal Special Topics

S

F

Fig. 1. Principle of force evaluation. Integrals are computed over Γ and Ω. Alternative paths(see Section 2.3 and figures therein) are separated by Δ.

on solid bodies, or even force density fields. We focus on flows around airfoils in thefollowing analysis, although the principles apply equally well to other geometries.Integral forces–such as lift and drag for aerodynamic flows–are traditionally mea-

sured directly using a force balance [2]. Since they are inherently intrusive, theyrequire calibration factors, and obtaining sectional forces is difficult at best. Lift mayalso be computed by integrating the surface pressure distribution, which can be mea-sured by pressure taps. However, both the surface pressure integration and the forcebalance approach become unfeasible if the Reynolds number Re and/or the modelitself are small.Non-intrusive, two- or even three-dimensional flow field measurements allow for a

different approach shown in Fig. 1: if both velocity and pressure on a surface enclosingthe airfoil are known, the integral momentum equation yields the force acting on saidairfoil. We apply and compare different methods of force evaluation in the first partof the paper. The main problem here is, that–to the best of the authors knowledge–there is no method for direct, non-intrusive measurement of the pressure field withina fluid. So one has to obtain it indirectly. If the distribution of the external bodyforce is known (or zero), the pressure field can be reconstructed from the measuredvelocity field.The second part of the paper will focus on the case where the external body force

itself is unknown. How can we calculate such a force density field, if the velocity fieldis given? This could be of interest for studying plasma actuators. They generate abody force which has been successfully applied to flow control problems, yet its spatialand temporal distribution is still debated [3].

2 Integral force

There are two classes of methods for evaluating the integral force a fluid exerts ona submerged body. The first class directly applies the integral momentum equationto calculate the force [4,5]. It explicitly requires the pressure on the surface, whichtherefore has to be determined in a preceding step. This step is crucial, since thepressure accounts for almost the complete lift and, for separated flows, most of thedrag.Several options have been proposed to reconstruct the pressure. In principle,

one can use the differential momentum equation to obtain the pressure gradient [4],then simply integrate it over the surface–a method prone to error propagation. Toimprove accuracy, a number of multi-path integration schemes have been applied.

Electromagnetic Flow Control in Metallurgy, Crystal Growth and Electrochemistry 93

Baur and Kongeter [6] implemented a method where four previously computed, neigh-boring nodes contribute to the pressure in a node. Oudheusden [7] uses a similarerosion scheme. Liu and Katz [8] take this even further and compute one node’s pres-sure from the average contribution of all radials connecting this node and the domainboundary. Another option is to solve the Poisson equation for the pressure [9–11]. Inregions of irrotational flow, it might be beneficial to apply Bernoulli’s equation [12].For the second class of methods, the pressure is eliminated via integral identities.

Noca et al. [13,14] proposed three variants: the “impulse equation,” the “momentumequation,” and the “flux equation.” The latter requires velocity information only ata surface, whereas the former two require the velocity in a volume. Wu et al. [15]offered an alternative and more compact formulation.Some of the previous studies compared results from a single method to reference

(force balance or wall-pressure) data. However, none compared the performance ofdifferent methods on a common set of data. In particular, Noca’s methods have notbeen applied widely.Here, we report the lift and drag coefficients obtained from optical velocity mea-

surements of separated flow over a flat plate at Re = 104 and an angle of attackα = 15◦. We compare several ways of evaluating the recorded images, namely PIVusing two interrogation window sizes, and Particle Tracking Velocimetry. On the dataset with the lowest divergence error we then compare the performance of four forcecalculation methods.

2.1 Methods

We consider an incompressible flow of a fluid of unit density. Typically applied fornumerical simulations, a force results from the integration of stresses over the body’ssurface S:

F = −∫S

(−pn+ τ ) ds, (1)

where p is the pressure, n the unit normal vector, and τ the shear stress. Alternatively,the force can be obtained from the momentum equation in integral form [4]:

FIM = − ddt

∫Ω

udV +

∮Γ

n · (−pI − uu+ T ) ds, (2)

applied to a control volume Ω, enclosing the body, which is bounded by a surface Γ.u denotes the velocity, T the viscous stress tensor, and I the identity matrix. Thedifferential momentum equation yields the pressure gradient,

∇p = −∂u∂t− (u · ∇)u+ ν∇2u, (3)

where ν is the kinematic viscosity. In principle, it can be directly integrated overthe surfaces S or Γ to give the pressure required in Eqs. (1) or (2), respectively. Werefer to this method as IM/DIFF in the following (Integral Momentum/DIFFerentialmomentum).Another option, referred to as IM/POIS, is take the divergence of (3), which leads

to a Poisson equation for the pressure:

∇2p = ∇ · (∇p) = −∇ · (u · ∇u− ν∇2u). (4)

Dirichlet boundary conditions could be feasible upstream or at far field boundarieswhere the flow can be considered irrotational, such that the pressure is known from


the Bernoulli equation or can be determined directly by integrating Eq. (3). However,neither approach is suitable for downstream or inner boundaries (S in Fig. 1, or anysurface inside Γ enclosing the body), so one has to resort to Neumann conditions. Forsimplicity and accuracy, we often use a rectangular inner boundary.Noca’s methods do not require the pressure explicitly. Similar to Eq. (2), the

“momentum equation” also includes a surface and a volume integral,

Fmom = − ddt

∫Ω

udV +

∮Γ

n · γmom ds, (5)

where

γmom =1

2u2I − uu− 1

k − 1u(x× ω) +1

k − 1ω(x× u)

− 1

k − 1[(x · ∂u∂t

)I − x∂u

∂t

]

+1

k − 1 [x · (∇ · T )I − x(∇ · T )] + T . (6)

We refer to this equation as MOM. Position and vorticity vectors are given byx and ω. The constant k = 2 if Eq. (5) is evaluated for planar data (yielding atwo-dimensional force), and k = 3 for volumetric data. We restrict ourselves to the2-D variants and evaluate planar data only.Noca et al. [14] also propose another two equations; however, for time-averaged

flow which we will focus on here, all of them become equal. Furthermore, the formu-las above originally involved terms to account for non-zero wall velocity, which areneglected here for simplicity.We implemented the methods in Python and thoroughly validated the code using

data from two reference cases computed by Direct Numerical Simulation (DNS). For2-D, quasi-steady circular cylinder flow at Re = 200, the absolute error of the liftand drag coefficients was less than 10−3. For 3-D, turbulent separated flow over aninclined flat plate at Re = 104, the error was on the order of 10−2 [16].

2.2 Experimental setup, image acquisition and evaluation

In an experiment corresponding to the latter DNS, we measured the velocity fieldby Particle Image and Particle Tracking Velocimetry (PTV). We placed a flat plate(chord 13 cm, thickness 8mm) with a rounded leading edge and a sharp trailing edgeat an angle α = 15◦ in a closed-loop water tunnel with a free-stream of nominallyU∞ = 8 cm/s, yielding Re = 10 400.As the force calculation requires the complete velocity field around the airfoil,

special care must be taken with the imaging setup. Oudheusden et al. [5] take ad-vantage of a symmetrical airfoil: they first capture, e.g., the pressure side flow at anangle of attack α, then set −α and capture the other. However, for non-symmetricairfoils, or if time-resolved forces are required, the full flow field must be captured.Two-sided illumination using a set of mirrors or a second light source will avoid anyshaded regions, but might be difficult to align properly. We use a translucent flatplate made from acrylic glass and mounted upside down (data has been flipped forvisualization). With one-sided illumination from below, this prevents any shadows inthe most interesting separation region. Diffraction at the leading and trailing edgesresults in only small shaded regions (c.f. Fig. 2) where the missing data is interpolatedby bivariate splines. If greater accuracy is required, the data could also be computedby numerical simulations fed with measured, unsteady boundary conditions [17].


Fig. 2. Divergence error of time-averaged flow, comparing different methods for computingthe velocity field: PIV with window size 16 × 16 and 8 × 8 px, PTV smoothed by 10 × 10Gaussian kernel, and raw PTV.

Our imaging setup consisted of a 250mW green diode pumped solid state laser(GMP532-300) as a light source and a Photron Fastcam 1024PCI 100K to record theimages. A thin (Δz ≈ 1mm) light sheet, formed by a cylindrical lens, continuouslyilluminated a plane extending in the direction of the flow (x) and normal to theflat plate (y). Covering approximately 20 × 20 cm, a total of 9980 images of 1024 ×1024 px were recorded at 60 frames per second. For the seeding, polyamide particles(Vestosint) of 25μm mean diameter were chosen. In the plane of the light sheet,x- and y-velocity components have been calculated from the images using PIVview-2C 2.4 from PIVTEC GmbH, Gottingen.We applied both PIV and PTV. Image pairs were dewarped first, then correlated

using multi-grid interrogation with final window sizes of 8× 8 or 16× 16 px and 50%overlap; image deformation and sub-pixel shifting were applied as well. For particletracking, PivTec’s ptv2c uses the PIV data in a predictor-corrector scheme to com-pute an individual particle’s displacement. We then time-averaged the displacementat each (integer) image location.

2.3 Results

All force calculation methods essentially originate from the integral momentum bal-ance, and Noca’s methods rely on a divergence-free flow to eliminate the pressure.Figure 2 therefore compares the divergence error ∂u/∂x+∂v/∂y of the time-averagedflow evaluated by PIV and PTV. Note that this figure is based on raw data, whereno interpolation of shaded regions has yet been applied.Aside from these regions of missing data (including the direct vicinity of the

flat plate), the maximum divergence is found in the separated shear layer and theseparated flow region at large. The PTV 16 × 16 px data fares best, followed by the8× 8 px data. As expected, the somewhat noisy, raw PTV data generates the largestdivergence errors. Gaussian smoothing with a 10 × 10 kernel reduces the error inregions of irrotational flow to approximately the same level as obtained from the PIV8×8 px data. However, for the separated flow region, the divergence error is still very


Fig. 3. Contours of pressure reconstructed using the Poisson equation. Underlying velocitydata from PIV (left) and DNS (right). The latter also includes the original pressure fromDNS (thin lines). The solid box indicates the inner boundary for the Poisson equation. Thedotted box marks the initial integration path used for the force reconstruction.

Fig. 4. Variation of the force coefficients with the distance Δ from the initial path shownin Fig. 3.

prominent. Increasing the kernel size to 20× 20 does not improve the results. Basedon these observations, we expect the PIV 16× 16 px data to yield the most accurateforce coefficients and focus on this case in the following.Figure 3 (left) shows the pressure calculated by the Poisson Eq. (4), in comparison

to DNS data (right). The wavy line near the profile in the PIV data stems from theunderlying Cartesian grid and the lack of data close to and inside the surface. On theright, both the original pressure from DNS is shown, as well as the pressure recon-structed from DNS velocity data. The original DNS pressure almost coinciding withits reconstruction indicates that for this time-averaged flow a purely two-dimensionalpressure reconstruction is valid. Any transversal components may obviously be ne-glected, including the non-vanishing out-of-plane Reynolds-stress terms u′w′, etc.Also, the PIV and DNS results are quite similar, although the PIV pressure mini-mum on the suction side is located further upstream. This is possibly due to the levelof free-stream turbulence in the experiments, which causes the separation bubble tobe shifted upstream. Free-stream turbulence is not modeled in the DNS. The PIV8× 8 px data (not shown) yields very similar results.Figure 4 compares lift and drag coefficients for different force calculation methods,

applied to the PIV 16×16 px data, and their variation when changing the integrationpath Γ (cf. Fig. 1). The initial path is shown in Fig. 3, corresponding to Δ = 0.


Table 1. Statistical analysis of force coefficients computed by different methods.

cL cDIM/ IM/ IM/ IM/DIFF POIS MOM DNS DIFF POIS MOM DNS

mean 1.12 1.13 1.02 1.02 0.21 0.27 0.16 0.314min−mean −0.17 −0.028 −0.074 −0.039 −0.037 −0.12max−mean 0.17 0.016 0.16 0.054 0.077 0.20max−min 0.34 0.044 0.24 0.093 0.11 0.32in % mean 30% 4% 23% 45% 42% 208%std dev 0.098 0.012 0.057 0.032 0.029 0.081

Negative Δ indicate only the upper boundary shifted inwards, therefore examiningthe effect of crossing the region of separated flow. For positive Δ, all boundarieshave been shifted outwards. For reference, we plotted DNS data obtained by Eq. (1).Unfortunately, no force balance data is available for this particular setup. Statisticsof the data from Fig. 4 are given in Table 1.Ideally, the coefficients would be constant and independent of the integration path.

However, a considerable variation with Δ is not uncommon, especially at large anglesof attack [5]. This can be used to assess the uncertainty of the force calculation. Wefind large errors for MOM for Δ < 0, most likely caused by the large divergenceerrors there. IM/DIFF is much less affected by Δ < 0. IM/POIS was not computedthere. Focusing on Δ > 0 and the PIV 16 × 16 px data, we find further differencesamong the methods. The lift coefficient calculated by IM/POIS is almost constant,but slightly above the reference. The curve for MOM shows some variations, yet stayswithin 10% of the reference, whereas IM/DIFF varies by up to 30% of its mean value.For the drag coefficient, all methods give values below the reference. IM/POIS againyields the largest value and is the most constant, varying by approx. ±0.05 for mostΔ. IM/DIFF calculates a mean drag coefficient about 0.1 less, with about the samestandard deviation and spread. MOM gives the lowest average drag, and the largeststandard deviation. We also conducted this study for the PIV 8 × 8 px data (notshown). While the mean curves are very similar, the standard deviation significantlyincreases.

2.4 Conclusions

We compared different image evaluation methods and different algorithms for calcu-lating the lift and drag forces exerted on an inclined airfoil. Particle Image Velocimetrywith a larger (16 × 16 px) interrogation window size results in the lowest divergenceerrors, and, subsequently, in the lowest standard deviation of the force coefficients.All methods require evaluation of line integrals along a path enclosing the airfoil.

In general, we find considerable variation with the integration path, except for thepressure Poisson method. Also, different methods give different results: the mean forcecoefficients vary by 10% (lift) and 60% (drag) among the methods. Noca’s momentumequation gives good results for the lift if applied outside the separated flow region.If that is not possible, direct integration of pressure should be used, which appearsto better tolerate divergence errors in that region. For the drag, however, Noca’smomentum equation yields the largest variations. Direct integration of pressure orthe pressure Poisson approach should be used instead.It is difficult to give exact figures for the accuracy since we lack direct force

measurements for comparison. Commercial force balance applications strive for anaccuracy of 0.1% [2]. This will hardly be achievable by PIV-based methods, as the


velocity measurement error itself is often quoted to be around 1% [1]. Accurate veloc-ity measurements are therefore crucial. Nevertheless, once the PIV data is available,the additional effort to apply force calculation methods is manageable and might yieldfurther insight. The reader is invited to contact the authors if he or she is interestedin the code.

3 Force density field

This second part is aimed towards obtaining a force density field from velocity fieldmeasurements. Such a method could support the understanding of plasma actuators,which have been a hot research topic in aerodynamics over the last decade due to theirpotential use for flow control applications. However, while their effectiveness has beenproven, their actual physics is still debated. There is no universally accepted model,let alone a rigorous derivation of the generated force from first principles. Directlymeasuring the integral force experimentally is straightforward, at least conceptually.However, obtaining the actual spatial force distribution is not; yet it is very importantnot only from a fundamental research perspective, but also for designing an actuatorfor a specific application.The force distribution could be computed directly from the differential momentum

equation if both the velocity and the pressure fields were known. But as the pressurefield cannot be measured non-intrusively, one has to work around this term somehow.Wilke [18] assumed that the pressure gradient term is much smaller than the bodyforce term itself and simply neglects the former altogether. Kotsonis et al. presented amethod [19] that requires accurate, time-resolved PIV over the initial transient periodafter the actuator has been switched on until a steady flow state is achieved. Weproposed a simple method [20] valid for the case of one force component being muchlarger than the other, and validated it experimentally using the known body forcedistribution of a Lorentz force actuator. If no component is negligible, the methodleads to a reconstructed force of equal curl. This might be sufficient to accuratelycapture the effect an actuator modeled that way would have on the velocity field,since only the curl of the body force contributes to production of vorticity, whereasthe remaining irrotational part “can often be balanced by pressure” [21].In the following, we present a method that yields both components of the instan-

taneous, solenoidal part of the force, but requires the pressure at the wall.

3.1 Method and numerical example

We introduce a stream function ansatz for the body force F = ∇ × Ψ + ∇Φ, i.e.,decomposing F into a solenoidal and an irrotational part. The x- and y-componentsof the solenoidal part are f = ∂Ψ/∂y and g = −∂Ψ/∂x, respectively. Substituting thisansatz in the incompressible, nondimensionalized Navier-Stokes equation and takingthe curl leads to a Laplace equation for Ψ,

∇2Ψ = ∇×(−dudt+N(u)− L(u)

).

whose right hand side merely requires velocities that can be measured by 2-D, two-component PIV. Note that the pressure gradient term has vanished since its curl isalways zero.Solving this equation requires boundary conditions. As sketched in Fig. 5,

sufficiently far from the actuator the force will have decayed to zero, hence ∂Ψ/∂y =∂Ψ/∂x = 0 and therefore Ψ = const on the far-field boundaries Γ2...4. Unfortunately,


Fig. 5. Left: original force vector field F , color-coded by magnitude. Right: absolute errormagnitude |F ′ − F |. Contour lines are separated by a factor of 10.

Fig. 6. Streamlines and contours of velocity magnitude. The box marks the region of forcing.

we can make no general assumption about the wall boundary condition on Γ1 forarbitrary actuators. However, if we know the wall pressure on Γ1, we can calculate ffrom the x-momentum equation there and apply a Neumann condition ∂Ψ/∂y = f .To test this method, we created a solenoidal vector field (basically, a velocity field

as produced by a flow solver) and use it as a prototype body force distribution. Thisforce drives a wall jet in a rectangular box of size (Δx,Δy) = (20, 10) bounded bysolid walls. From an initially quiescent fluid, we calculated a steady-state solution(see Fig. 6), then sampled the velocity field on Ω with a resolution of 128× 64 nodes,typical for an average PIV setup. We submitted the data, along with the pressureaccordingly sampled at the wall, to the above procedure implemented in Python andcomputed a force field F ′. Shown in Fig. 5 (right), the error |F ′ −F | peaks near thewall, but is less than 10−2 for most of the domain. Further increasing the resolutionsignificantly reduces the error (not shown).

3.2 Conclusions

Unless we know the complete pressure field, it is, in general, not possible to recover theirrotational part of the force. Also, if we knew the complete pressure field, computingthe body force from the momentum equation would be straightforward. Since thesequantities are unknown, we have shown that the most appropriate method, bothsimple and robust, uses only the solenoidal part. So the next best thing we cando is to go with the solenoidal part, only, for which we have shown an easy, robust


method. We have validated this method using synthetic, numerical data and obtainedsufficiently accurate results. For an experimental validation, we would need combinedPIV and wall pressure measurements of the flow created by an actuator of knownforce distribution. In principle, this is possible using Lorentz force actuators.With further advances in pressure measurement techniques (e.g., pressure sensitive

paint), this method could potentially be used to recover the solenoidal part of thebody force induced by plasma actuators.

Support by Deutsche Forschungsgemeinschaft as part of the Collaborative Research CentreSFB 609 (projects C1 and C2) is gratefully acknowledged.

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Deriving forces from 2D velocity field measurements