The Structure of Urban Flows

Understanding the flow in urban environments is an increasingly relevant problem due to 1 its significant impact on air quality and thermal effects in cities worldwide. In this review we provide 2 an overview of efforts based on experiments and simulations to gain insight into this complex physical 3 phenomenon. We highlight the relevance of coherent structures in urban flows, which are responsible 4 for the pollutant-dispersion and thermal fields in the city. We also suggest a more widespread use of 5 data-driven methods to characterize flow structures as a way to further understand the dynamics of 6 urban flows, with the aim of tackling the important sustainability challenges associated to them. 7

configured to record three-dimensional data. Note that the testing zone was selected to be highly 258 gusted, precisely because the objective was to assess the different methodologies used in pedestrian 259 distress studies. The full-scale testing was aimed at obtaining data on mean-flow speed. In addition, 260 the data was also used to help characterising the flow. The measurements were then compared with 261 the results of wind-tunnel experiments and numerical simulations.  The data-acquisition system was composed of 700 sonic anemometers equipped with a TR90-T probe. 277 Those were installed at half-height distance from the top of the block. In addition, acrylic plates with 278 pressure taps were also installed in the northwestern and southeastern faces of the blocks. Using the 279 aforementioned setup two datasets were acquired. From the analytical perspective, the study rests on 280 two major axes. On the one hand, the study of approaching flow conditions, characterised through 281 the statistical descriptions of turbulent flows. On the other hand, the study revolved around the 282 relation between pressure difference and velocity. The authors examined the probability distribution of the velocity at the horizontal wind direction by means of the streamwise velocity magnitude, the 284 standard deviation as well as the velocity range in both southeastern and northwestern winds. As far 285 as the relation between pressure and velocity is concerned, it was examined by means of the pressure 286 coefficient, obtained for every position using the least-squares method.    determine the mean wind speed; although these are not the most accurate sensors for this type of 312 measurement, their accuracy was considered to be sufficient their study. The main advantage lies on 313 their omnidirectionality, which eases the installation process since no realignment is needed. Secondly, 314 multi-hole probes such as Cobra probes, were used to measure the incoming wind speed on the top 315 of the model. Multi-hole probes are typically used in high-resolution measurements of turbulent 316 flows. However, those probes are limited by their insensitivity in flows slower than two meters per 317 second as well as their directionality. The third measurement system was hot-wire anemometry, which 318 overcomes the limitations of the two aforementioned system. Nevertheless, hot-wire anemometers 319 exhibit limitations in terms of spatial resolution and sensitivity to wind direction [63]. To summarize, 320 Irwin probes and hot-wire anemometers were used independently to measure the flow at pedestrian 321 level, i.e. within the model. In addition, a Cobra probe was used to obtain data of the incoming wind    Other studies apply this kind of experiment to concrete applications. In the present work,      comes from the use of a planar PIV. We know that urban flows are complex and three-dimensional. 443 In this way, the use of a planar measurement system will inevitably introduce errors. The authors 444 used numerical data to evaluate the magnitude of the error, concluding that the uncertainty remained   On the other hand, the wind-tunnel data was within the standard-deviation range found in the      antisymmetric parts of the tensor (also known as the strain-rate and rotation tensors, respectively).

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The Q criterion was used by Krajnovic and Davidson [134] to identify the vortical structures around 687 a wall-mounted cube; note that this technique has the advantage of correctly identifying vortices 688 even under high shear. Another popular method for vortex identification is the so-called λ 2 criterion 689 by Jeong and Hussain [18], which is based on analyzing the eigenvalues of S 2 + R 2 . In particular, 690 they define a vortex core as the connected area where S 2 + R 2 has two negative eigenvalues. If the 691 eigenvalues are defined as λ 1 ≥ λ 2 ≥ λ 3 , the vortex core can be identified as the region where λ 2 < 0.

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There are other vortex-identification methods based on the velocity gradient, such as the ∆ criterion by  Besides the previously mentioned methods, which are based on the velocity gradient, there is another approach which is based on an integral quantity instead: the normalized angular momentum Γ 1 proposed by Graftieaux et al. [137], which is defined as follows: has paid special attention to novel data-driven techniques to perform modal decompositions of the flow.

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Combining the acquired physical knowledge of these techniques with new machine-learning strategies it is possible to create fast and efficient reduced-order models (ROMs) accurately modelling the flow its corresponding eigenvalues and eigenvectors will be real numbers, and the eigenvectors will be 783 orthogonal among them, forming an orthogonal basis of modes.

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The classical POD method solves the following eigenvalue problem: where the optimal orthogonal modes Φ j (POD modes) that better approximate a dynamic field, are related with the largest eigenvalues λ j . In three-dimensional turbulent flows, this method is prohibitively expensive since it is based on solving the eigenvalue problem of the covariance of a state vector changing in time, with dimensions proportional to the spatial degrees of freedom (J × J). Hence, the snapshot method should be used instead. This method is based on the fact that the most energetic POD modes are the same as those obtained with the transposed covariance matrix as : The dimensions of C is proportional to the snapshot number from eq. (A1), i.e. K × K (let us remember

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SVD is a type of factorization that captures the directions of a matrix in which vectors can either grow or shrink. These directions are given by the eigenvalues and eigenvectors of a rectangular matrix. Similar to the snapshot method, SVD applied to the snapshot matrix (A1), decomposes the flow field into spatial (W) and temporal (T) modes and singular values Σ, as: where W W = T T = I, with I being the N × N unit matrix, and Σ is a diagonal matrix composed 790 by the singular values (σ 1 , · · · , σ K ), which are ranked in decreasing order. This method is strongly 791 connected to the previous eigenvalue problems, where σ 2 j correspond to λ j , the columns of W are the POD modes Φ j (ranked consistently with their corresponding eigenvalues) and the columns of T are 793 the temporal modes Ψ j .

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The number of SVD (or POD) modes most relevant to describe the flow can be identified using several criteria. Several methodologies are described in Ref. [163], where SVD and POD methods are also known as principal component analysis (PCA). When solving fluid dynamics problems the standard SVD error, estimated for a certain tolerance ε 1 , defines the most relevant modes in POD analyses based on the singular values as: The most relevant modes can then be used to provide low-dimensional approximations of complex 795 flows.

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The good performance of POD/SVD algorithms has been shown in the identification of coherent for k = 1, · · · , K, where u m (x, y, z) are the DMD modes weighted with the amplitudes a m , ω m are their corresponding frequencies and δ m are their associated growth rates, representing the temporal growth or decay of the modes in time. The DMD algorithm introduced by Schmid [150] is based on the linear relationship between two consecutive snapshots via the linear Koopman operator R. Starting from the snapshot matrix (A1), where the snapshots are organized equidistant in time with time step ∆t, the DMD method is defined using the following Koopman assumption: where the matrices V K−1 1 and V K 2 contain from the first to the last but one snapshots, and from the second to the last snapshots of the data base, respectively. The Koopman matrix R contains the dynamics of the system, while the DMD frequencies ω m and growth rates δ m are the computed eigenvalues and the computed eigenvectors are used to construct DMD modes u m (x, y, z). Recently, Le Clainche & Vega [155] introduced the higher-order dynamic mode decomposition (HODMD) method, which is an extension of DMD for the analysis of complex flows (turbulence, multi-scale, flows in transitional regime,...) and noisy experimental data. This method combines the Koopman assumption (A8) with the Taknes' delayed embedded theorem [171]. Hence, HODMD relates d time-delayed snapshots (sub-matrices) using the high-order Koopman assumption as: which can be understood as the window-shift process carried out in power-spectral density (PSD).

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Finally, as in DMD, the dynamics of the system is contained in the several Koopman operators, 827 R 1 , · · · , R d , the eigenvalues of which represent the DMD frequencies and growth rates and their 828 eigenvectors are used to construct the DMD modes, as in standard DMD.

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The HODMD algorithm proceeds in two steps (see more details in Ref. [155]):

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• Step 1: dimension reduction. Applying truncated singular value decomposition (SVD) to the snapshot matrix V K 1 yields: The matrixT K 1 is the dimension-reduced snapshot matrix. The number of SVD modes retained 831 in this approximation N is defined as the spatial complexity. These modes are selected as in the 832 SVD algorithm presented in §A.1. A (tunable) tolerance ε 1 estimates the standard SVD error, as 833 described in eq. (A6).

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• Step 2: the DMD-d algorithm for the dimension-reduced snapshots. The high-order Koopman assumption is applied to the reduced snapshot matrix, resulting in: After some calculations, the several Koopman operators R 1 , · · · , R K are grouped into a single matrix, the eigenvalue problem of which provides the DMD modes, frequencies and growth rates that define the DMD expansion (A7). This expansion is sorted in decreasing order of the mode amplitudes and it is further truncated by eliminating the modes such that: for some tunable parameter ε 2 . The number of modes retained in this expansion, M, is called as 835 the spectral complexity.
When d = 1 in equation (A11), HODMD reduces to standard DMD, defined as: while for d > 1, HODMD can be understood as the result of applying standard DMD to a set of 837 enlarged snapshots, defined by the delayed snapshots. Hence, HODMD combines the advantages of 838 standard DMD with some consequences of the delayed-embedding theorem by Takens [171].