A novel approach for extracting the coherentstructure of the turbulent flow field in

atmospheric boundary layer using UAV swarmflights

M. Kiaee, T. Auerswald*, J. Bange

Center for Applied Geo-scienceEnvironmental Physics Group

University of Tuebingen

email: torsten.auerswald@uni-tuebingen.de

June 22, 2014

1. Introduction1 With the rapid progress of available computation resources, performing large scale computa-tional fluid dynamics (CFD) simulations in atmospheric boundary layer (ABL) is becoming moreattractive. Since Smagorinsky invented the large eddy simulation (LES) based on the idea fromKolmogorov, it has become one of the prominent approaches in CFD simulations for large scalephenomenon. Every CFD simulation requires the proper spatial and temporal inlet conditionsof the flow in order to kick off a realistic simulation. Since LES does not explicitly resolve theviscous sub-layer of the turbulent flow, it is fairly reasonable to import the large structures ofthe flow as the inlet instead of the full resolution flow field (Huang (2009) ). Furthermore, byconsidering the various turbulence generators for feeding the LES (e.g. Auerswald (2012) ) theCS generation could be a missing piece in the chain of measurement-simulation. Moreover, it iswell discussed that the Lagrangian coherent structure (CS) is expected to be responsible for contri-bution of high energy transfer within ABL neglecting the small energy cascades (Steiner (2011)). Quantification of such contribution to momentum and heat fluxes is necessary to interpretmeasurements in the ABL. The idea behind the CS extraction is to develop and implement amethod in order to capture the coherent structures of the turbulent flow. The extraction is per-formed by using the eigenvectors of the flow field. These eigenvectors or the so called properorthogonal decomposition (POD) of the turbulent field are used to create an orthonormal bases ofthe turbulent field (Lumley (1994) ). Considering the orthogonality property of the eigenvalue,they are employed as a complementary tool for the linear stochastic estimation (LSE) method(Adrian (1994) and Druault (2004) ). LSE is responsible for the reconstruction of the origi-nal turbulent field. The current work by implementing the common CS extraction of engineeringfluid mechanics (e.g. Sun (2012) ) creates a novel approach in the boundary layer meteorology.The tremendous field size of the ABL makes it almost impossible to create a high spatial resolutionof the flow field. Consequently the original ABL field is unknown. In this work, the latter problemis treated by using the large eddy simulation (LES) results mimicking the original field. Othercomputational cost obstacle rises in the POD method in which the eigenvalue problem should besolved. Hence, calculation of the eigenvectors of the large matrices resulting in the CS extractionis a computational demanding task. (Sharma (2012) ) used the QR algorithm in order perform

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the principal component analysis (PCA). As a result the the computation cost reduced from the14dn2 for singular value decomposition (SVD) method to 2dn2 + 2dth for the QR algorithm (d isthe dimensionality of feature space and n is the number of training feature vectors and t is therank of data covariance matrix and h is the dimensionality of reduced feature space). Since PODand PCA are similar methods, the QR is also expected to reduce computation time for the CSextraction tremendously. From the measurement point of view, the recent improvements of un-manned aerial vehicles (UAV) made important impacts on ABL meteorology measurements (e.g.Bange (2006) ). Having the measurements from the UAVs in hand the established method coulduse the data to enhance the measurements by appropriate stochastic inter and extrapolations.

Furthermore, This paper intends to emphasize the role of the so called swarm flight of theUAVs. In such flight several UAVs fly simultaneously in parallel lines with specified distances andgather temporal and spatial turbulent field information. These obtained information from theUAV swarm flights are in perfect harmony with the offered approach in this paper. The technicalcomplexity of performing such a flight which is currently in progress at environmental physicsgroup at university of Tuebingen (ref Norman??) and is outside of the scope of this paper.

2. MethodsFrom the basic theories of turbulence modeling, it is common that flow velocity field is decomposedinto a mean and a fluctuating part. The fluctuating part itself could be decomposed into a coherentand an incoherent terms

u = u + u′ and u′ = ucohe + uinco (1)

in which ucohe is the part representing the large structures of the turbulent flow.

2.1. Linear Stochastic Estimation (LSE)Before Adrian’s LSE is explained some definitions and explanations are necessary.Definition1 In probability theory, a conditional expectation is the expected value of a real randomvariable X with respect to a conditional probability distribution Y and is shown as X|Y .Definition 2 Stochastic Estimation is approximation or estimation of a random variable in termsof some other random variables which are known.Definition 3 Adrian’s Conditional Eddy is the best mean square estimate of the estimated velocityfield u′ given data e is the conditional expectation of u′ given e, u′|e. (Adrian 1974) It is shownthat under the condition that elements of u′ and e are joint normally distributed then u′|e is alinear function of e. Then one could write the linear stochastic estimation as follows

u′ = linear estimate of u′|e =M∑

i=1Li · e (2)

where M is the number of event data and each Li is a vector. u′ estimated in this way is calledconditional eddy. The estimation coefficients vectors Li are chosen so that the mean square errorof the conditional expectation minimized (Adrian 1990 ). This will require the orthogonality ofthe error (u′|e −

∑i Li · e) to the data itself (e) in an average sense

(u′|e −∑

i

Li · e) · e = 0 (3)

1The bold upper and lower case symbols stand for matrices and vectors respectively. The normal letters standfor scalars. e.g. R is a matrix, Ri = r is a vector and Rij = R or r is a scalar. In the context of continuousfunctions and variables both are treated with the normal letters. e.g. F (x) and f(X) are functions of variables xand X respectively.

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This will lead to the Y ule − Walker equations, an M × M system of linear equations for thecoefficients Lij .

M∑j=1

ejek Lij = eku′i (4)

The left hand side coefficients for Lij of the previous equation is the correlation of event data(known) and right hand side the estimated correlation. Challenges at this point:

• The field is only known at some discrete points along some specific lines.

• Before further analysis the 3d field in space should be reconstructed.

As a remedy an estimation of the field should be performed by means of covariance estimation.Adrian’s LSE can be used for accomplishing this goal. Following the (Delville (1994) ) notationit can be written

u′(x′, t) =M∑

j=1L(x′) · u(x, t) + O[u2(x, t)] (5)

Following Adrian’s proof result in eq. (4) and by minimizing the error the following equation canbe written R(x1, x1) . . . R(x1, xN )

.... . .

...R(xN , x1) . . . R(xN , xN )

L1(x′

i)...

LN (x′i)

=

R(x1, x′1) . . . R(x1, x′

M )...

. . ....

R(xN , x′1) . . . R(xN , x′

M )

(6)

Considering that the left hand side of the previous equation is a constant matrix, R, dependingon the event data

R L(x′) = R(x′)

this can be expressed in other terms as follows

L(x′) = R−1R(x′) (7)

By considering that matrix R should be non-singular (Correlation vector of each point to allpoints should be independent from correlation of the other points to all points), calculation of Lrequires R(x′). The solution is M (estimated space size) linear system of equations each of size N(event data size). The main concern is that whether the following expression for extrapolation issatisfied using linear, or even higher order stochastic estimates where the time scales are concerned(Delville (2006) )

uEstimated(x′) ?= uExact(x′) (8)

This is generally the case in truly isotropic and homogeneous turbulence when the unconditionalgrid is a subset of conditional grid (Adrian 1980; Bonnet et al. 1994; Glauser et al. 2004) or where the spatial separations between the two grids are small relative to large length-scalestructure, δx < L. Similarly Adrian (1996) showed that the linear estimate of a random variablecould be exact only for separations between the unconditional and conditional terms that weresmaller than Taylor micro-scale.

2.2 Proper Orthogonal Decomposition (POD)In order to understand Lumley’s POD it is necessary to mention some definition here.

Definition 1 Following Lumley’s definition of empirical eigenfunctions, one way to approachto the best functional fit to a statistical data is to envision the ensemble member u as vector in afunction space. Then we wish to find a fixed vector ϕ as nearly parallel as possible to u in somestatistical sense. Let us define the projection of u on ϕ as (u, ϕ). Making ϕ as similar as possible to

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u then means to maximize the (u, ϕ). One way of increasing the (u, ϕ) is to increase ϕ to infinity.In order to avoid this we should instead maximize the normalized functional

Ψ = (u, ϕ)√(ϕ, ϕ)

(9)

In which Ψ is a deterministic functional. If we do not introduce any statistics the ϕ which maximizethe Ψ will be u itself which will not help. This problem is solved by maximizing the mean square(averaging and multiplying Ψ by its conjugate) in the following equation instead.

λ = E(u, ϕ)(u, ϕ)∗(ϕ, ϕ)

≥ 0

By making use of the Schwartz kernel theorem and by doing some statistic operations previousequation will be simplified as the following equation

(R(x, x′), ϕ∗(x′))) = λϕ(x) (10)

in which R(x, x′) is the covariance function. If R(x, x′) is an integrable function (which is in ourcase) we can simplify the equation in real space in a more familiar integral equation form of∫

R(x, x′)ϕ(x′)dt′ = λϕ(x) (11)

There are infinite number of solutions , ϕi(x), corespondent to λi to this equation and are calledempirical eigenfunctions and eigenvalues.

Definition 2 Linear combination of empirical eigenfunction of the flow ϕ(1), . . . , ϕ(N) sortedby the magnitude of the corresponding eigenvalue λ(k) ≥ 0 defines coherent structure of the flow.Following this definition, the eq. (1) can be rewritten as

u′(x, t) =∑

i

ai(t)ϕ(i)(x) + uinco(x, t) (12)

Obviously by finding the appropriate coefficients set ai the coherent structure of the flow isrealized. In the reverse approach, projecting (dot product (., .) in function space) the estimatedvelocity field onto the space related eigenfunctions.

ak(t) =∫

Ωuest(x, t) ϕ(k)(x)dx (13)

Eq. (13) requires the knowledge about the estimated velocity field uest(x, t). This is how LSEdemands POD as a complementary tool.

uest(x, t) = L u(xref1 , . . . , xrefN, t) (14)

2.3 Solving the POD equationIn order to find the empirical eigenfunctions of the flow from (11) and having a continuous functionspace the integral eigenvalue problem (EVP) for the function R(x, x′) with first n eigenfunctionsas ϕ(k)(x) we can write 1

∫Ω

3∑q=1

Rpq(x, x′)ϕ(k)q (x′)dx′ = λ(k)ϕ(k)

p (x) ; k ∈ 1 . . . n and x ∈ Ω (15)

1Equivalent of j, the space index, can be used for writing xj = x0 + j∆x and is different than p and q, whichare the covariance indexes

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Which is an Integral Eigenvalue Problem (IEVP). In order to be able to solve the IEVP numericallyit should be converted to a discrete EVP format. The covariance function of velocity in spacex = (x, y, z)T is

R(x, x′) = R(x, y, z, δx, δy, δz) =

u(x, t)u(x′, t) u(x, t)v(x′, t) u(x, t)w(x′, t)v(x, t)u(x′, t) v(x, t)v(x′, t) v(x, t)w(x′, t)w(x, t)u(x′, t) w(x, t)v(x′, t) w(x, t)w(x′, t)

(16)

Then the left hand side of the IEVP can be rewritten as∫Ω

3∑q=1

Rpq(x, x′)ϕ(k)q (x′)dx′ =

N∑i=1

3∑q=1

w(xi)Rpq(xi, xj)ϕq(k)(xj) = λ(k)ϕ(k)

p (xi) (17)

By choosing an appropriate integration scheme, B = wR, the previous IEVP is simplified intoa standard EVP. Weight vector w depends on the integration scheme and the discretization size∆x. For a equidistant grid and using the simple rectangle discretization this can be condensedinto one 3N × 3N B matrix as follows

B = ∆x

Rpq(X1, X1) . . . Rpq(X1, XN )...

. . ....

Rpq(XN , X1) . . . Rpq(XN , XN )

and p, q = 1, 2, 3 (18)

The equation (17) can now be represented in finite (discrete) dimension space eigenvalue problemfor n × n matrix B with n eigenvectors ϕ(k)B11 . . . B1n

.... . .

...Bn1 . . . Bnn

ϕ1

...ϕn

= λ

ϕ1...

ϕn

or B ϕ(k) = λ(k)ϕ(k) (19)

For a fast calculation of eigenvectors of the B, matrix QR decomposition method is being used.For the m × n matrix B the QR procedure is as follows

B = B0 = Q0R0 (20)

In which Qi is an orthogonal and Ri is an upper diagonal matrix. If Qi and Ri are multiplied inthe reverse order and this procedure continues

Bi+1 = RiQi (21)

And then Bi+1 is decomposed to the new Qi+1 and Ri+1 that

Bi+1 = Qi+1Ri+1 (22)

It can be provedlim

i→∞Qi = I (23)

Multiplying all Qi in process ends up to matrix ΦB with eigenvectors of B in its columns.

ΦB = limi→∞

Q0 . . . QN (24)

It can also be proved that RN will contain the eigenvalues of B on its diagonal

ΛB = RN or limi→∞

diag(Ri) = λ(1), . . . , λ(n) (25)

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2.4. Estimation of the covariance functionThere are two cases of estimation for the covariance function, Interpolation and extrapolation.Each of these requires an their own specific treatment. The trivial nature of an extrapolationmethod reduces the certainty about the results as the temporal or spatial distances from the eventrises. However, One important condition which should be always certainly fulfilled is the boundarycondition (which might also be located far from the event). To satisfy the boundary conditionin both inter and extrapolation scheme the eigenvectors of the POD are used as a guess. Eacheigenvector of the POD itself satisfy the boundary condition hence the linear combination of themalso represent this characteristic.

1. Interpolation of Rij : For interpolation a summation of polynomials and trigonometric func-tions are being used.

Rij(x, x′) =NGC∑k=1

Qk(sin Pk) (26)

Pk = P k00 + P k

20(x − x0)2 + P k11(x − x0)(x′ − x0) + P k

02(x′ − x0)2

Qk = Qk20(x − x0)2 + Qk

11(x − x0)(x′ − x0) + Qk02(x′ − x0)2

2. Extrapolation of Rij : For extrapolation, the polynomials part of the eq. (26) dominatesthe whole series for large separation and is not well suited for this purpose. Extrapolatingempirical eigenfunctions and then Orthogonality of eigenfunctions∫

ϕ(i)(x)ϕ(j)(x)dx = 0∫

ϕ(j)(x)ϕ(j)(x)dx = 1 (27)

From before the POD equation is the the following IEVP∫R(x, x′)ϕ(n)(x′)dx′ = λ(n)ϕ(n)(x) (28)

Using the eq. (27) and by multiplying both sides of eq. (28) by ϕ(n)(x) and integrating, allterms will be zero except∫ ∫

R(x, x′)ϕ(n)(x′)ϕ(n)(x)dx′dx = λ(n)∫

(ϕ(n)(x))2dx (29)

Eq. (29) shows an important relation which indicates that each ϕ(i)(x)ϕ(i)(x′) is a basis forR(x, x′) with the coefficients λ(i). This expression in other term can be written as

R(x, x′) =N∑

n=1λ(n)ϕ(n)(x)ϕ(n)(x′) (30)

As an overview the overall procedure of CS extraction is abstracted as follows:1. Series of the velocity field vector either comes from the UAV measurements. (e)

2. Two-points correlation tensor in known velocity discrete xyz space. (R) using eq. (16).

3. Empirical eigenfunctions (Coherent Structures) and eigenvalues (ϕ(k) and λ(k)) using eq.(17).

4. Estimation of continuous correlation tensor (Rest) using eq. (26).

5. Linear stochastic estimation operator from Rest (Lij) using eq. (7).

6. Estimated velocity random field is being calculated (uest) using eq. (5).

7. Estimated random field is projected onto the eigenfunctions (ak) using eq. (13).

8. Turbulent field is reconstructed using eigenfunctions and their respective instantaneous co-efficients. (a1ϕ(1) + a2ϕ(2) + . . . )

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Figure 1: The projection of the turbulent field onto POD eigenfunctions (Left) and the originalturbulent field (Right)

3. Test caseIn order to apply the discussed method to ABL, a very fine resolution time series of turbulentfield (original field) and a reduced information of the original field (coarse field) are required. Theoriginal field is required for the validation procedure. As it has been highlighted capturing theoriginal field by purely measurement is not feasible for very large volumes such as ABL. However,an alternative possible approach is to use the fine resolution result of LES as the reference. Havingthis original field in hand, it is possible to generate the coarser information by picking out somearbitrary nodes. For this test case, the simulation result is from PALM software from universityof Hanover which is an atmospheric boundary layer flow solver (PALM (2014) ). The grid sizeof the sample simulation is 129 × 129 × 66. By rewriting the eq. (9) in the more familiar format of

u(X) = a1ϕ1(X) + a2ϕ2(X) + a3ϕ3(X) + . . . (31)

and by using the orthogonality of eigenfunctions∫X

u(X)ϕ1(X)dX = a1

∫X

ϕ21(X)dX (32)

The coefficients can be calculated as

ai =∫

Xu(X)ϕi(X)dX∫X

ϕ2i (X)dX

(33)

The effect of using a few number of eigenvector can be observed in figure [1]. The left field showsonly the first few eigenvectors which are summed with appropriate coefficients ai.

Then the linear stochastic estimation is applied to find the lr coefficients in following equation

R(x, xREFi ) =

Nr∑r=1

lr(x)R(xREFr , xREF

i ) (34)

knowing the li then the following expression can be used to fined uest

uest =Nr∑r=1

lru(xREFr ) (35)

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This is the estimated velocity field using the extrapolated POD eigenfields. Using five lines ofthe measurement information as the swarm flights measurements replacement, the original field isconstructed. The comparison between the original and the estimation of part of the original fieldcan be seen in figures [2,3].

Figure 2: Original turbulent field Figure 3: Estimation using POD+LSE

4. ConclusionThis paper offers an approach to find the coherent structure of the ABL by using UAV measure-ment. By combination of QR algorithm with the proper orthogonal decomposition, a complemen-tary tool has been used for the linear stochastic estimation. Implementing this complementarytechnique it is possible to reconstruct the ABL flow field in large scales. This provides a tool tocreate the proper inlet for LES. The current approach is validated by assuming the LES resultsto be the original field. This provides the basis for further analysis in the real world measure-ment in which the original field is unknown. The current paper approach is specifically developedto be part of the measurement technique called the UAV swarm flights. By combination of theUAV swarm flights and the results of this paper an "optimized measurement in the atmosphericboundary layer by means of simulation (OATS)" will become possible.

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