Dimension reduction of thermo-fluid mechanisms in irradiated particle-laden turbulence

Center for Turbulence Research Annual Research Briefs 2019 Dimension reduction of thermo-fluid mechanisms in irradiated particle-laden turbulence By L. Jofre, Z. R. del Rosario AND G. Iaccarino 1. Motivation and objectives Dimensional analysis provides fundamental understanding of a physical system through the detailed examination of its units. The underlying principle of the approach is based on the notion of similarity, which postulates that relationships between physical quantities do not vary if the measurement units are changed. This central result implies that simpler small-scale experiments can be utilized to study larger-scale phenomena. In addition, one major advantage is that dimensional analysis typically yields a smaller number of independent dimensionless variables than the original measured quantities. Hence, the dimensionality of the system is reduced, and as a result fewer experiments are needed to characterize its response, i.e., quantity of interest (QoI), to a set of inputs. 1.1. Irradiated particle-laden turbulence The investigation of thermal radiation interacting with particle-laden turbulent flow is of great importance in a wide range of scientific and engineering problems. For instance, phenomena reminiscent of irradiated particle-laden turbulence are ubiquitous in the fields of earth and combustion sciences, such as the impact of preferential concentration on the rate of droplet coalescence and evaporation in atmospheric clouds (Shaw, 2003; Dodd & Jofre, 2019) and the fluid mechanics of reacting boundary layers and plumes (Tieszen, 2001; Jofre & Urzay, 2017). Of particular interest to this work is the study of these physical processes in the context of volumetric particle-based receivers for energy harvesting (Ho, 2017) in concentrated solar power (CSP) systems. Inertial particles in homogeneous isotropic turbulence (HIT) exhibit complex interactions between the phases as a result of preferential concentration and turbulence modulation (Balachandar & Eaton, 2010). Preferential concentration is the mechanism by which heavy particles tend to avoid intense vortical motions and accumulate in regions of high strain rate, whereas turbulence modulation refers to the alteration of fluid flow characteristics in the near-field region of particle clusters as a result of two-way coupling effects. The physical complexity is further increased by the simple addition of walls. In that case, turbophoresis becomes an important mechanism for augmenting the spatial inhomogeneity of the dispersed phase by driving particle accumulation at the walls. Consequently, the analysis and characterization of particle-laden turbulent flow are challenging endeavors; many experimental and computational research studies have been devoted to this objective over the past decades, e.g., Caporaloni et al., (1975); Squires & Eaton, (1991); Dunton et al., (2017); Jofre et al., (2019). In addition to particle-flow coupling, the problem studied in this work involves an additional layer of complexity by considering the heat transfer from the particles to the fluid via radiation absorption. The engineering application motivating the understanding of these phenomena is the improvement of energy harvesting in volumetric particle-based solar receivers. This innovative technology is expected to increase the performance of CSP plants by avoiding the 167 Jofre, del Rosario & Iaccarino necessity of heat-exchanging stages. However, the physical mechanisms governing these systems are still not fully comprehended. Examples of recent work focusing on this problem include the interaction between radiation, particles, and buoyancy in HIT (Zamansky et al., 2014), the impact of heating on the settling of particles (Frankel et al., 2016), the effect of Stokes number and polydispersity on particle-gas heat transfer rates (Pouransari & Mani, 2017; Rahmani et al., 2018), and the quantification of uncertainties and sensitivity analysis of complex systems (Jofre et al., 2017; Masquelet et al., 2017; Fairbanks et al., 2020; Jofre et al., 2019). 1.2. Physical basis of dimensional analysis Dimensional analysis offers a general framework for reducing complex physical systems to a simpler form prior to obtaining a quantitative answer. Central to its basis is the concept of similarity (Cantwell, 2002). In physical terms, similarity refers to some equivalence between two phenomena that are quantitatively different. In mathematical form, similarity refers to a transformation that preserves some property, implying that a smaller number of variables is needed to explain the phenomenon at hand. For example, under particular conditions there is a direct relationship between the movement of large masses of air in the atmosphere and the motion of a fluid in a small-scale laboratory (or computational) model. The challenges are to find (i) those conditions and (ii) the transformation between them; in this case, the same ratio of inertial to viscous forces, i.e., Reynolds number. Dimensional analysis aims to help solve these two challenges in general problems by providing a set of mathematical techniques and methodologies. The premise of this scientific-technical discipline is that the form of any physics-based description of a system, e.g., conservation equations and experimental correlations, must be such that the relationship between the actual physical quantities remains valid independently of the magnitudes of the base units utilized. This feature provides a number of very useful outcomes in terms of (i) facilitating the inference of similarity laws, (ii) producing a basis for out-of-scale modeling, (iii) providing support for dimensionality reduction approaches, and (iv) obtaining insight that is independent of the system of units utilized. However, it presents some limitations. For example, (i) an incomplete, or unnecessary, set of independent variables may complicate the analysis (del Rosario et al., 2019), (ii) the framework is not robust to external simplifying assumptions, (iii) the set of scale-free relations obtained is not unique, and (iv) there is no formal approach to quantify the relative importance between dimensionless groups. In this regard, this work proposes a data-driven methodology aimed at addressing the last two shortfalls by means of augmenting Buckingham’s πtheorem with ideas developed in the fields of ridge functions (Pinkus, 2015) and active subspaces (Constantine, 2015). In addition, the physical interpretability of the results is enhanced by introducing a linear algebra approach to re-express the dimensionless groups on a user-selected basis that can be combined with inspectional analysis of the conservation equations. 1.3. Objectives and organization of the work As previously introduced, the exploration and analysis of complex systems, especially multiphysics flow problems, can be systematically approached by considering the important dimensionless groups characterizing the relations between the underlying physics phenomena. Extraction of the dimensionless parameters is also very useful for engineering practice as it allows one to identify important directions in the input space for the efficient design and optimization of systems. Therefore, the objectives of this work are (i) to present a semi-empirical methodology, based on the seminal work by Constantine et al., 168 Dimensionality reduction of irradiated particle-laden turbulence (2017), to effectively infer important dimensionless groups from data obtained by means of computational (or laboratory) flow experiments, and (ii) to utilize the methodology to characterize important dimensionless groups in irradiated particle-laden turbulence. Detailed descriptions of the physics modeling and computational approach utilized in this work to study irradiated particle-laden turbulence are presented by Jofre et al., (2019) and Esmaily et al., (2015, 2018), respectively. The remainder of the paper, therefore, is organized as follows. A detailed presentation of the data-driven dimensional analysis methodology is given in Section 2. In Section 3, the configuration of the model problem is described in terms of physics, setup, and system parameters. Next, in Section 4, results and their analysis are discussed. Finally, in Section 5, the work is concluded and future directions are proposed. 2. Data-driven dimensional analysis In this section, we describe the integration of classical dimensional analysis with modern dimension reduction techniques. The resulting tools enable data-driven discovery of relevant dimensionless numbers, while accounting for the practical realities of large-scale simulations. The following subsections detail requisite background material. 2.1. Dimensional analysis and the πsubspace To enable data-driven dimensional analysis, we first connect classical techniques to a modern subspace reduction perspective. Dimensional analysis is a classical dimension reduction technique (Buckingham, 1914). Its central result is the Buckingham πtheorem: Given a set of dimensional inputs z∈Rndthat predict a dimensionless QoI, i.e., Q= f(z), the functional relationship may be re-expressed in terms of a smaller number of dimensionless numbers π∈Rnpvia π=ψ(π1, . . . , πnp). The set of valid dimensionless inputs π={π1, . . . , πnp}can be determined from the dimension matrix D. Following the notation of del Rosario et al., (2019), let d(·) be a vectorization of the dimension function (Barenblatt, 1996). Then, the dimension matrix for zis given by D= [d(z1),...,d(znd)].(2.1) Valid dimensionless numbers can be formed by products of the inputs as πi= nd Y j=1 zvij i,(2.2) with the vectors {vj}np j=1 satisfying Dvj= 0. In this formulation, the Buckingham π theorem can be understood in terms of the rank-nullity theorem. In other words, the number of independent dimensionless numbers — with independence defined by the usual notion of vector independence applied to vi— is given by np= dim(R(D)) −dim(N(D)),(2.3) where dim(·) is the subspace dimension, R(·) denotes the range, and N(·) denotes the nullspace. The Buckingham πtheorem is silent on the choice of a basis for the nullspace of D. The choice of appropriate dimensionless numbers is often a matter of experience. However, in this work we use data to inform a useful selection of relevant πgroups. To connect Buckingham’s πtheorem to subspace reduction, we make the following observation. Select a set of nominal conditions z0∈Rndand define xi= log(zi/zi,0) for 169 Jofre, del Rosario & Iaccarino i= 1, . . . , nd. Under this transform, we can write π=ψ(exp(v> 1x1+ log(π1,0)),...,exp(v> npxnp+ log(πnp,0))), =ψ0(V>x+ log(π0)),(2.4) where πi,0are the dimensionless numbers evaluated at the constant nominal conditions zi,0,π0= [π1,0, . . . , πnp,0]>, and the composition exponentiation is collapsed within ψ0 to highlight mathematical structure. Intuitively, this structure exhibits variability only within a subspace of its input domain, and is invariant to orthogonal perturbations. From Eq. (2.4), one can show that variation in the QoI occurs only through variations within R(V); since this object is derived from Buckingham’s πtheorem, it is called the πsubspace (del Rosario et al., 2019). Note that the QoI must be non-dimensionalized for this property to hold. To elucidate the importance of the πsubspace, let R(V)⊥be the orthogonal complement of R(V), and let y∈ R(V)⊥. Then π(x+y) = ψ0(V>(x+y)), =ψ0(V>x+ 0), =π(x), (2.5) making precise the previous statement of invariance. This invariance shows that Eq. (2.4) is a ridge function in its inputs x(Pinkus, 2015). A restatement of this invariance is that the gradient of πis constrained. In mathematical form, note that ∇xπ=V∇ξψ0(ξ),(2.6) and as a result the following relation is satisfied ∇xπ∈ R(V).(2.7) This observation will prove useful as we discuss active subspaces below. Note that while Buckingham’s πtheorem defines a set of valid dimensionless numbers, it does not provide a specific choice of an appropriate basis V(Constantine et al., 2017). The function ψ0 may have additional structure with further invariance properties. These properties can be expressed in terms of active subspaces. 2.2. Active subspaces and dimensional analysis The active subspace is a dimension reduction concept introduced by Russi, (2010) and developed by Constantine, (2015). Let f(x) be some differentiable QoI on a domain with integral weight ρ(x)∈R≥0. The active subspace is then defined in terms of the matrix C≡Z∇xf∇xf>ρ(x)dx.(2.8) Since Cis by construction symmetric semi-positive definite, it admits an eigenvalue decomposition of the form C=UΛU>. The eigenvalues need to be sorted first in decreasing order as λ1≥ ··· ≥ λnd. Next, a threshold separating {λ1, . . . , λnd}into large {λ1, . . . , λnA}and small {λnA+1, . . . , λnd}eigenvalues is defined to generate the split U= [UA,UI]. The final result is that the directions uiare then ordered in decreasing order of importance with respect to variation in the QoI f, made quantitative by the eigenvalues. The active subspace is then given by R(UA), where the columns of UAform a basis for the subspace. In the case where frepresents a physical equation respecting dimensional homogeneity, 170 Dimensionality reduction of irradiated particle-laden turbulence Buckingham’s πtheorem guarantees the range constraint in Eq. (2.7). From this fact, it is clear that if w∈ R(V)⊥, we have w>Cw =E[w>∇xf∇xf>w] = 0.(2.9) An immediate consequence of this nullity is that if one requires the active subspace to include only those directions for which λi>0, then R(UA)⊆ R(V).(2.10) In words, the active subspace — computed along the transformed variables x— is a subset of the πsubspace. Furthermore, using the vector entries in Eq. (2.2), the active directions uA,i can be directly interpreted as dimensionless groups. The selection of the active subspace dimension nArequires some caution. If f(x) = g(V>x) is a ridge function in its inputs x, then one can use Eq. (2.5) to show that λi= 0 for i > dim(R(V)); thus, nA≤dim(R(V)). However, one may also make a pragmatic choice of nAbased on preserving a user-defined level of variance in the function (Lee, 2019). We illustrate navigating these challenges in the subsections below. 2.3. Using data-driven dimensional analysis Using active subspaces together with dimensional analysis enables a number of investigative approaches. The active directions can be used to produce a summary plot of the input-to-output response (Cook, 2009). A summary plot consists of a plot of the response fiagainst reduced coordinates ξA≡U> Axi. In the case where dim(UA) = 1, this can be easily visualized as a scatter plot, while dim(UA) = 2 can be plotted with additional difficulty as a value-colored scatter plot. Even when the active subspace is too high dimensional to directly visualize, the active directions UAcan be directly re-interpreted by re-expressing them in terms of a usersupplied basis of interpretable dimensionless numbers. Data-driven dimensional analysis identifies πgroups which are relevant to the data at hand, but the resulting products of inputs need not involve simple powers. Conversely, the common dimensionless numbers, such as the Reynolds number, Mach number, and Prandtl number, carry physical interpretation; generally, ratios of competing effects. Re-expressing a given active direction uin a user-defined basis Vof these interpretable dimensionless numbers is useful for physically interpreting the results. In practice, if one solves a linear system for weights wvia Vw =u,(2.11) then, the dimensionless groups represented by ucan be re-expressed as π= exp u> dx, = exp w>V>x, = exp w1(v> 1x) + ···+wnp(v> npx), =πw1 1×···×πwnp np. (2.12) This allows one to construct data-driven dimensionless groups from a user-selected basis of standard dimensionless numbers, greatly aiding in physical interpretation. 2.4. Approximating and evaluating active subspaces Studying active subspaces is further complicated by the realities of approximation. Generally, Eq. (2.8) cannot be computed exactly. Direct approximation of Eq. (2.8) requires 171 Jofre, del Rosario & Iaccarino evaluation, or approximation, of the gradient ∇xf; finite differences increase the computational complexity of simulation by a factor d+ 1. In our target application, this corresponds to an order-of-magnitude increase. Rather than approximating the matrix in Eq. (2.8) directly, we instead leverage recent algorithms using variable projection (Hokanson & Constantine, 2018). Conceptually, this procedure embeds a least-squares polynomial approximation within an optimization over the Grassmann manifold. The need for gradient data ∇xfis ameliorated by assuming a polynomial model form and target dimensionality, allowing gradient approximation based on the assumed model. In this way, one may use point evaluations f(xi) in place of gradient samples ∇xfi, greatly reducing computational expense. 3. Description of the irradiated particle-laden turbulence system The setup of the problem is inspired by the study of thermo-fluid mechanisms in volumetric particle-based solar receivers. The analysis of this type of system involves the interaction of particles, turbulence and radiative heat transfer. A complete description of the problem setup and system parameters is presented in the subsections below. 3.1. Problem setup The study of particle-laden turbulence in an irradiated environment is performed by considering two domains. An isothermal cube of size Wis utilized as a particle-laden HIT flow generator in which the fluid phase (initial density ρ0,f and temperature T0,f , constant dynamic viscosity µf) is volumetrically forced (Bassenne et al., 2016). The dispersed phase is initialized at the same time and temperature as the fluid with Np,0 monodisperse particles (constant density ρpand diameter dp) randomly distributed in the volume. The turbulence forcing scheme is targeted to produce an averaged turbulent kinetic energy, k∞,f , such that the ratio between domain size and Kolmogorov length scale, η, is W/η ∼ O(102), and therefore the small-scale features of the flow are not significantly affected by the periodic boundaries. This first domain is designed to provide turbulent steady-state fluid-particle inflow conditions to the (second) radiated section. The rectangular radiated section is of size L×W×Win the streamwise (x, in/outflow boundaries), spanwise (y, periodic boundaries) and crossflow (z, periodic boundaries) directions, respectively. The turbulent fluid-particle flow mixture is sampled in time from a yz-plane in the HIT volume and introduced to this second domain by adding a bulk velocity U0to the streamwise velocity component. To achieve similar turbulence characteristics as in wall-bounded flows, the ratio between root-mean-square velocity fluctuations, urms, in the HIT domain and U0is selected to be urms/U0∼ O(10−1), and the gravitational acceleration is not considered as its effects are negligible compared to the inertia of the bulk flow. As the fluid-particle mixture flows through the domain, it is irradiated with uniform intensity I0. The result is that particles (constant isochoric heat capacity Cv,p and absorption coefficient p) absorb thermal radiation, increasing their temperature, Tp, and subsequently transferring energy to the surrounding fluid (constant thermal conductivity λfand isobaric heat capacity CP,f ) by thermal exchange (constant fluid-particle heat convection coefficient h). The fluid-particle mixture in this problem is optically thin, allowing us to model the radiation absorption by particles with an algebraic model; in other words, all particles receive the same amount of radiation intensity. As a result of the particles heating and transferring thermal energy to the carrier fluid, the average fluid temperature Tfin172 Dimensionality reduction of irradiated particle-laden turbulence Parameter Value Parameter Value W[0.038 : 0.042] m CP,f /Cv,f 1.4 (diatomic ideal gases) L[0.152 : 0.168] m h[1 ·103: 1 ·104] W/(m2·K) T0[285 : 315] K Np,0[9.5·105,6: 10.5·105,6] U0[1 : 5] m/sρp[1 ·103: 1 ·104] kg/m3 k∞,f [0.1 : 0.5] m2/s2dp[1 ·10−6: 1 ·10−5] m ρf,0[0.5 : 1.5] kg/m3Cv,p [1 ·102: 1 ·103] J/(kg ·K) µf[1 ·10−5: 2 ·10−5] Pa ·sp[0.25 : 0.75] λf[1 ·10−2: 3 ·10−2] W/(m ·K) I0[9.5·105: 10.5·105] W/m2 CP,f [1 ·103: 2 ·103] J/(kg ·K) Table 1. List of independent input parameters and their range of values. creases along the streamwise direction. This deposition of energy accelerates the flow by means of thermal expansion due to a decrease in fluid density ρf. 3.2. System parameters The study conducted in this work is designed with the objective of mimicking an experiment as it would be carried out in a laboratory facility. Following this approach, the system is characterized by 16 input parameters that can be varied independently to collect data. The list of input parameters and their range of values are described in Table 1. The ranges of W,L,T0, and I0are obtained by adding 5% to/subtracting 5% from their nominal values as these are parameters that in a laboratory facility would not be easily modified in large proportions. The intervals for ρf,0,µf,λf,CP,f ,h,ρp,Cv,p, and pare based on engineering values for material properties taken from Poling et al., (2001); the generic fluid is considered to be a diatomic ideal gas resulting in CP,f /Cv,f = 1.4. The levels of U0and k∞,f are designed together to obtain realistic ratios of fluctuating-to-bulk velocity in turbulent wall-bounded flows (Pope, 2000). Finally, the ranges of dpand Np,0 are selected to study micron-sized particles representative of conditions in volumetric particle-based solar receivers at relatively small (case I: Np,0= 1 ·106±5%) and large (case II: Np,0= 1 ·107±5%) particle number densities, np,0. Data are collected by computing PP-DNS of the problem for different values of the input parameters sampled from a randomized Halton sequence. A total of 256 samples have been computed for I and II (128 per case). The range of values for W,L, and k∞,f are utilized to define, following the estimations described by Pope, (2000), the mesh resolution required to perform the calculations such that the significant turbulent scales are captured. The resulting Eulerian meshes for the HIT and radiated domains correspond to uniform Cartesian grids of sizes 512 ×512 ×512 and 2048 ×512 ×512, respectively. The time-averaging of the QoIs is computed by taking the ensemble average of 15 flow-through times (FTTs), defined as FTT ≈L/U0, on yz-planes — the solution is symmetric in the yand zdirections — after the first thermal transient FTT is surpassed. 173 Jofre, del Rosario & Iaccarino 4. Results and discussion This section presents and analyzes the data acquired by computing the set of samples described in Section 3, and provides a discussion of the results obtained from the methodology introduced in Section 2. 4.1. Data-driven inference of principal πgroups Focusing on the normalized increment of fluid temperature Q≡(Tf−T0)/T0, the datadriven polynomial ridge approximation strategy described in Section 2 is utilized to infer principal dimensionless groups from the data collected. This methodology requires one to approximate Qby a ridge function with a multivariate polynomial gof dimension np and total degree din the form Q≈g(V>x), where xare the log-transformed inputs. By construction, npimposes the number of subspace dimensions (equivalent to dimensionless groups) to be inferred, whereas ddetermines the degrees of freedom (polynomial order) available to fit the data along each dimension. The adequate npand dvalues are not known a priori. In consequence, a study based on the coefficient of determination (CoD) is carried out to obtain suitable values for these two parameters as a first step of the methodology. The CoD, typically denoted as R2, provides a measure of how well observed outcomes are replicated by a model relative to the proportion of total variation of outcomes explained by it. Given a dataset of n= 128 values y1,...yn, each associated with a fitted, or modeled/predicted, value f1,...fnfrom which a residual ei=yi−fican be computed, the mathematical definition of R2is written as R2≡1−SSres SStot ,(4.1) where SSres =Pn i=1(yi−fi)2=Pn i=1 e2 iis the residual sum of squares, and SStot = Pn i=1(yi−¯y)2is the total sum of squares (proportional to the variance of the data) with ¯y=n−1Pn i=1 yithe mean of the observed data. Following the statistical definition introduced above, the R2analysis is summarized in Figure 1 by considering Qat the outlet of I and II. The subspace dimensionality of Q slightly increases along the streamwise direction as more radiative energy is absorbed by the system. Thus, selecting npand dfor the data at the outlet will provide a polynomial model with enough dimensions and fitting coefficients to be safely applied to infer dimensionless groups at different stations in the domain. The set of subspace dimensions and polynomial degree combinations is constrained by the amount of data available. Particularly, as shown in Figure 1, the maximum number of subspace dimensions is limited to 6, and the highest polynomial degree is restricted to 4. The cross-validation is performed by splitting the data into 8 groups of same size, and the results are represented by means of boxplots. As depicted in the figure, restricting the polynomial gto 1 dimension is not sufficient for approximating the data, especially for II, for which R2presents large variability skewed toward small values. In contrast, considering 3 dimensions complicates the analysis, while it does not significantly increase R2with respect to np= 2. Therefore, the pragmatic number of subspace dimensions for I and II is 2, constructed by utilizing polynomial degrees equal to 3 and 4, respectively. This selection results in R2values presenting small variability and close to 1. As introduced in Section 2, the projection weights v1and v2correspond, respectively, to the exponents of the input parameters composing each of the two principal dimensionless groups, π1and π2, approximated by the polynomial ridge function. The labeling of these two groups has been arranged such that π1is the subspace most aligned with the 174 Dimensionality reduction of irradiated particle-laden turbulence Figure 1. Coefficient of determination, R2, results of different subspace dimension and post-projection polynomial-fit order models for Q≡(Tf−T0)/T0at the outlet for I (a) and II (b). The grayscale color scheme indicates the polynomial order of the model. principal direction obtained from approximating the data with a one-dimensional polynomial ridge function. The Vvalues are very similar for I and II. This result indicates that the two cases investigated exhibit equivalent behavior in terms of important dimensionless groups, and consequently the analysis below discusses them together by considering I. The R2results depicted in Figure 1 show that the first subspace dimension accounts for almost 80% of the variation in the data. Focusing on the first subspace dimension, the data-driven inferred expression for the dimensionless group π1for Q≡(Tf−T0)/T0 at the outlet of the radiated section is given by π1=W−0.510 ×L0.612 ×T−0.414 0×U−0.174 0×k−0.002 ∞,f ×ρ−0.104 f,0×µ0.001 f×λ−0.010 f(4.2) ×C−0.161 P,f ×h0.013 ×N0.045 p,0×ρ−0.146 p×d0.002 p×C−0.095 v,p ×0.210 p×I0.224 0. Analogously, the inferred definition for π2is π2=W−0.084 ×L−0.491 ×T−0.422 0×U−0.069 0×k0.027 ∞,f ×ρ−0.097 f,0×µ−0.074 f×λ0.053 f(4.3) ×C−0.076 P,f ×h0.007 ×N0.478 p,0×ρ0.113 p×d0.535 p×C−0.032 v,p ×0.084 p×I0.101 0. These expressions provide a quantitative decomposition of the principal πgroups as a function of the independent (input) parameters of the problem. A more insightful, and easier to interpret, decomposition in terms of standard fluid mechanics dimensionless numbers is discussed in the next subsection. The input-to-output response of Qat the outlet of the radiated domain as a function of the πgroups inferred by the ridge function polynomial is depicted in Figure 2. The fit obtained by the first subspace dimension shown in Figure 2(a) reveals that Qincreases monotonically with π1, and presents exponential growth for π1>1. In addition, the twodimensional summary plot depicted in Figure 2(b) qualitatively indicates that Qgenerally increases also with π2. However, as shown by the plot, the importance of this second dimensionless group is significantly small with respect to π1as most of the variation in the data (grayscale color gradient) is captured by the latter. 4.2. Interpretation in terms of standard dimensionless numbers The data-driven methodology described in Section 2 allows one to easily re-express the πgroups as powers of standard dimensionless numbers, Eq. (2.12), through the simple 175