DNS of a turbulent boundary layer using inflow conditions derived from 4D-PTV data

In the present study, unsteady PTV data from a state-of-the-art PTV measurement are used as an inflow boundary condition in a DNS. To evaluate the applicability of such a hybrid experimental-numerical approach, the canonical case of a flat plate TBL at a momentum thickness Reynolds number of \(Re_{\theta }{\approx }\text {4000}\) is examined, where \(Re_\theta {=}{\bar{\rho }}\,{\bar{u}}_e\theta /{\bar{\mu }}_e\). Here, \(\theta \) is the incompressible momentum thickness, the index e denotes values evaluated at the edge of the boundary layer and overlined variables \(({\bar{\square }})\) represent time-averaged values. The external data are coupled via a relatively straightforward proportional forcing scheme (see Sect. 5.2) in a 3D inlet subdomain, downstream of which the flow is allowed to evolve freely. The current work is unique in that it uses 3D, unsteady PTV data from a state-of-the-art PTV technique rather than a single 2D plane, an achievement made realizable thanks to the availability of such datasets in the first place. Hybrid studies typically focus on the enhancement of experimental data within the complete area of interest, such as determination of the pressure field or the reduction of experimental noise. The current study focuses on the implementation and evaluation of a realistic turbulent inflow condition in which the incoming flow is already at a developed turbulent state, quantified by \(Re_{\theta }\), though not spectrally complete. For reference, conventional DNS studies of flat plate TBLs often require significant computational effort to arrive at an equivalent \(Re_{\theta }\). By qualitatively evaluating the resulting flow field’s agreement with the input of the original EFD contribution, a first estimation of the potential of the methodology is assessed. Additionally, through analysis of the spectral development of the TBL within the DNS, inferences can be made about the degree of spectral deficiency inherent to the measurement and pre-processing chain. To the knowledge of the authors, the present study is the first of its kind in terms of resolution and dimensionality of inflow.

The present paper addresses the primary question of whether a 3D DNS using coupled 4D-PTV data is possible and worthwhile. Since the utility of the coupling as it pertains to the definition of an upstream BC is of central focus, the success of the study is evaluated through comparison of the freely evolving DNS field with the experimental field. This evaluation will reveal the net effect of factors such as potential mismatches in the exact far field condition, for example the freestream pressure gradient and presence of periodic span-wise conditions, on the evolution of the TBL flow. The central question of how the limited resolution of the external data source, in the present case meaning the finite tracer density, affects the development of turbulent kinetic energy k and the arrival upon a state of constant k production will be addressed. In short, the limiting factors of the methodology are to be investigated, such as resolution criteria.

The paper is organized according to the following structure. Section 2 provides a compact summary of the experiment from which the data for the simulation were derived. Section 3 characterizes the experimental data directly. Section 4 presents the interpolation method used to assimilate the Lagrangian PTV data to the rectilinear DNS grid. Section 5 presents the numerical method used, including information about the flow solver NS3D, the applied boundary conditions, as well as the hybrid coupling scheme used to introduce velocity perturbations into the flow domain. The results of the simulation itself are shown in Sect. 6, focusing on the mean flow (6.1) and wavenumber domain analysis (6.2). The simulation and experiment are then directly compared in Sect. 7, including snapshots of the instantaneous flow fields in Sect. 7.1, a quick demonstration of the application of spatial filtering in Sect. 7.2 and a look at the correlation between the experimental and DNS in Sect. 7.3. Finally, Sect. 8 summarizes the results presented in Sects. 6 and 7 and discusses the conclusions of the study.

Bin-averaging is performed to derive averaged flow metrics in a stationary 3D reference frame. Binning is the process of quantizing or aggregating non-uniformly distributed discrete data into ranges. Here, binning is necessary for deriving a spatiotemporal average of PTV data in the form of discrete ‘point cloud’ style tracer measurements with sufficient statistical confidence. Spatial bin boundaries are defined, and tracer measurements for all times residing in each bin are averaged. Figure 3 shows the near-ZPG region as well as the stream-wise and span-wise binning schemes. Additionally, a superposition of the eventual DNS domain boundaries is shown. In the wall-normal direction, a bin width of 200 µm was used. Figure 2a shows the wall-normal \({\bar{u}}^{+}\) and \(\overline{u^{\prime }u^{\prime }}^{+}\) profiles for the \([n_{x} {\times } n_{z}]\) = \([5 {\times } 1]\) binning scheme, where it is evident that the profiles consistently collapse when non-dimensionalized in wall coordinates. Primes \(({\square }^{\prime })\) denote fluctuations with respect to the Reynolds averaged mean value. Figure 2b shows the same wall-normal profiles for the \([n_{x} {\times } n_{z}]\) = \([1 {\times } 4]\) binning scheme, showing a degree of span-wise non-uniformity present within the experimental data due to the presence of the upstream bubble rakes mentioned in (Schanz et al. 2019). For reference, Fig. 2 contains ZPG profiles from (Eitel-Amor et al. 2014) for comparable \(Re_{\theta }\), represented by gray lines. Additionally, the \(u^{+}{=}y^{+}\) trend for the viscous sublayer, \(u^{+}{=} 1 / \kappa \cdot \mathrm {log}(y^{+})+C\) (with \(\kappa {=}0.41\), \(C{=}5.2\)) trend in the logarithmic layer and an approximation for the transition region from (Spalding 1961) are plotted.

In an effort to characterize the datasets received, one metric of interest is the nearest neighbor (NN) distance \(r_{\text {NN}}\) of every tracer at every timestep. A k-dimensional tree based method was used to establish the distance between every combination of roughly 550K tracers per timestep at each of 1359 timesteps. The minimum of these distances, the nearest-neighbor distance (or radius), is then saved as an additional scalar which can be averaged in time and space. Additionally, the standard deviation of \(r_{\text {NN}}\) for tracers at all times in a given spatial bin delivers information about the uniformity of the tracer density. Figure 4 shows that, on average, the mean tracer distance \({\bar{r}}_{\text {NN}}\) in the ‘well-seeded’ part of the experimental flow is roughly 5.3 mm. The plot colors indicate the x-bin (see Fig. 3), demonstrating the consistency of the seeding density in the stream-wise direction in addition to the wall-normal y direction between 100 and 2000 dimensionless viscous wall distance units \(y^{+}{=}yu_\tau /\nu _{w}\). Additionally, the standard deviation of \(r_{\text {NN}}\) is roughly 1.8 mm.

Close to the wall (\(y^{+}{<}100\)), both the mean \(r_{\text {NN}}\) and its variance increase. Since no new tracers are ‘randomly walking’ from underneath the wall, the tracer replenishment is only coming from above (elsewhere from above and below). When tracer bubbles hit the wall they burst, so for a given random walk there are more sinks than sources of tracers. This phenomenon is well illustrated in Fig. 4, where the mean NN radius \({\bar{r}}_{\text {NN}}\) starts to change at \({\bar{r}}_{\text {NN}}\) from the wall, which is where the effect of the wall acting as a tracer ‘sink’ would be expected to first become visible on average, being that for \(y{<}{\bar{r}}_{\text {NN}}\) a sink exists within the local sphere defined by \({\bar{r}}_{\text {NN}}\). Also, the standard deviation NN radius has a sharp increase also at \({\bar{r}}_{\text {NN}}{=}5.3\ \text {[mm]}\), which is likely due to tracer bubbles bursting. When this happens, the sudden elimination of one tracer from the local population causes a jump in \({\bar{r}}_{\text {NN}}\) between timesteps, hence increasing the variance of a time-collapsed population.

The experimental data are in the form of sampled tracer trajectories, meaning that the velocity of the turbulent flow field is only known at scattered point locations for a given timestep. Since the coupled hybrid region acting as an unsteady inlet will require information at a set of stationary points (rectilinear grid points) a spatial interpolation of the data is necessary. Interpolation of sampled, limited-resolution data will generally introduce some level of error with respect to the continuous (and unknown) ‘ground-truth’ data, in this case the turbulent flow in the laboratory. The choice of interpolation method is therefore important and in the present case was chosen with two priorities in mind, namely the minimization of interpolation error and the fulfillment of known conservation relationships such as mass conservation. Naturally, these priorities are aligned in that reduction of error implies fulfillment of conservation laws (and vice-versa). For this reason, significant effort was invested into the comparison and validation of potential methods.

Another class of commonly used interpolation methods for tasks of this type is radial basis function (RBF) based interpolation. Practical details of RBFs and their application to interpolation are available in (Schaback 2007; De Marchi and Perracchione 2018) and a formal mathematical description can be found in Buhmann (2000, 2003). Similar to polynomial and the special case of piecewise polynomial (spline) interpolation, RBF interpolation involves setting up and solving a system of linear equation in the classic form \(A\lambda {=}f\), such that a coefficient vector \(\lambda {=}A^{-1}f\) is obtained and used to form a continuous interpolant through linear dot-product summation of the coefficients and a set of basis functions (e.g., \([1,x,x^{2},x^{3},\dots ]\) in 1D polynomial interpolation). RBF interpolation differs from classical polynomial interpolation in that the interpolant is constructed from the set of distances between the known point coordinates rather than the spatial coordinates themselves. This has enormous advantages when extending the method to higher dimensions, where polynomial interpolation quickly becomes expensive, difficult, and is not guaranteed to produce an interpolant for any arbitrary set of points when using a given set of basis functions (Mairhuber 1956). Discussion on the differences between polynomial interpolation and RBF interpolation, especially concerning interpolation in multiple dimensions can be found in (Buhmann and Jäger 2019). Section 4.2 gives a summary of radial basis function interpolation for 3D scalar fields as well as the extension to a ‘divergence-free’ formulation for 3D vector fields.

In its conventional formulation, RBF interpolation seeks to derive a continuous approximation (or interpolant) function s of a scalar-valued function f: \(s({\mathbf {x}}) {\approx } f({\mathbf {x}})\). Here, the scalar-valued function f is assumed to be a function of (and is interpolated in) 3D space \({\mathbb {R}}^{3}\), though the method is equivalent for functions of any space \({\mathbb {R}}^{n}\) since it transforms the support point coordinates to a set of shifted scalar distances. The function f is therefore a multivariate function of three coordinate dimensions, or equivalently the coordinate vector \({\mathbf {x}}{=}[x,y,z]\). If f is known at N known points in space, the condition is set on s that the input f must be exactly recovered at these N points, i.e., \(s({\mathbf {x}}_{i}) {\equiv } f({\mathbf {x}}_{i})\) for \(i{=}1,2, \dots , N\). The interpolation process has two main steps: the solution of the scaling vector \(\lambda \) and the formation and evaluation of the interpolant. In the first step, a linear system of equations (LSE) of the form \(A \varvec{\lambda }{=}f\) is set up: where the A matrix is the square (\(N {\times } N\)) distance matrix with every Euclidean distance \(r{=}\Vert {\mathbf {x}}_{i}-{\mathbf {x}}_{j}\Vert \) element evaluated by a scalar-valued, positive-definite RBF \(\phi \). Typical candidates for \(\phi \) include \(\phi {=} r\), \(r^{3}\), \(r^{2}\text {log}(r)\) or \(e^{-\varepsilon r}\), where \(\varepsilon \) is an adjustment parameter. The f vector contains the values of \(f({\mathbf {x}}_{i})\) at the N known points. The \(\lambda \) vector is the unknown scaling coefficient vector, which can be solved for through inversion of A as \(\varvec{\lambda }{=}A^{-1} f\). In the second step, the interpolant function \(s({\mathbf {x}})\) is built and evaluated in the form \(s {=} B \lambda {=} B A^{-1} f\), where the B matrixis an M\({\times }\)N matrix similar to A, but is now the RBF evaluation of the distance matrix between the N known and M ‘new’ points. The s vector is the resulting vector of interpolated values at the M new points. In the present case, the number of tracers at a given timestep within and in the vicinity of the inlet region fluctuates around the average value of \(N{\approx }6000\). The tracer data are used to form the interpolant, which interpolates the PTV flow field to \(M{=}[33{\times }424{\times }1280]{=}17.9\text {M}\) grid point locations for use in the DNS.

In the present study, the interpolated scalar f would represent each individual component of the vector-valued velocity field \({\mathbf {u}}{=}[u,v,w]^{\text {T}}\) and the ‘new’ points would be the inlet grid points. Interpolating each velocity component independently would not leverage known physical relationships between these quantities, such as being approximately divergence-free \(\nabla {\cdot } {\mathbf {u}}{\approx }0\) due to mass continuity and the effectively constant mass density \(\rho \) at the low experimental freestream Mach number of \(M_{\infty ,\text {exp}}{=}0.019\). Therefore, a vector field \({\mathbf {f}}({\mathbf {x}})\) can be interpolated using a matrix-valued RBF kernel (Narcowich and Ward 1994). A significant advantage of this formulation is that the RBF kernel can be formulated in such a way that the resulting interpolated vector field inherits the quality of being divergence-free by virtue of how the RBF kernel is constructed. Formal mathematical foundations and analysis of this method are found in Lowitzsch (2002, 2005) and further applied examples are presented in McNally (2011); Yang et al. (2014). This extension reduces interpolation error by eliminating \(\nabla {\cdot }{\mathbf {u}}{\ne }0\) error. Alternatively considered, the accuracy of the interpolant is increased because it now has more detailed constraints.

A vector-valued approximation function \({\mathbf {s}}({\mathbf {x}})\) of a vector-valued field \({\mathbf {f}}({\mathbf {x}})\) is now once again sought such that \({\mathbf {s}}({\mathbf {x}}) {\approx } {\mathbf {f}}({\mathbf {x}})\). Drawing on Helmholtz’s theorem, it is known that any smooth, well enough behaved vector field \({\mathbf {f}}\) can be decomposed as the sum of a curl-free (irrotational) and a divergence-free (solenoidal) field \({\mathbf {f}}{=}\nabla p {+} \nabla {\times } {\mathbf {g}}\), where the gradient of the scalar potential \(\nabla p\) is curl-free by identity (\(\nabla {\times } (\nabla p){=}0\)) and the curl of the vector potential \(\nabla {\times } {\mathbf {g}}\) is divergence-free by identity (\(\nabla {\cdot } (\nabla {\times } {\mathbf {g}}){=}0\)). In the special case that the vector field \({\mathbf {f}}\) is known a priori to be divergence-free (\(\nabla {\cdot }{\mathbf {f}}{\equiv }0\)), then it can be shown that Helmholtz’s theorem reduces to \({\mathbf {f}} {=} \nabla {\times } {\mathbf {g}}\) (Griffiths 2014), i.e., that \({\mathbf {f}}\) is the curl of a vector potential field \({\mathbf {g}}\). The vector function \({\mathbf {f}}\) can still not be uniquely defined as the curl of a vector field \({\mathbf {g}}\), i.e., \({\mathbf {f}}{\equiv }\nabla {\times } {\mathbf {g}}\) because for any \({\mathbf {g}}\), where \(\nabla {\times } {\mathbf {g}}{=}{\mathbf {f}}\), the offset of \({\mathbf {g}}\) by the gradient of any arbitrary scalar potential field \(\nabla p^{\prime }\) would also yield a field where \(\nabla {\times } ({\mathbf {g}}{+}\nabla p^{\prime })\) is also \({=}{\mathbf {f}}\) (see (Jackson 2002)). This non-uniqueness is summarized by the concept ‘gauge freedom’ or ‘gauge invariance,’ i.e., that inconsequential additional degrees of freedom (realized by ‘gauge transformations’) are available which yield equivalent observed states of a system (Jackson and Okun 2001). To break the gauge invariance and hence the non-uniqueness problem, additional conditions are needed, i.e., the gauge must be ‘fixed.’ One common choice of gauge is the Coulomb gauge, equivalent to saying that the vector potential \({\mathbf {g}}\) is itself divergence-free, i.e., \(\nabla {\cdot }{\mathbf {g}}{=}0\), which can be implemented by setting \({\mathbf {g}}{=}\nabla {\times }{\mathbf {h}}\) (see (Jackson 2002; Stewart 2003)). Thus \({\mathbf {f}}\) can be uniquely defined as \({\mathbf {f}}{\equiv }\nabla {\times }(\nabla {\times }{\mathbf {h}}){=}\nabla (\nabla {\cdot }{\mathbf {h}}){-}\nabla ^{2}{\mathbf {h}}\). This is the starting point for the definition of the divergence-free approximation function \({\mathbf {s}}\), which can be expanded in 3D Cartesian coordinates:The 1D derivative operator \(\frac{\partial }{\partial x_{i}}\) is shortened to \(\partial _{x_{i}}\) above for compactness. The operator \(\{-\Delta {\mathbf {I}}{+}\nabla \nabla ^{T} \}\) maps the field \({\mathbf {s}}\) to \({\mathbf {h}}\) and is also used in the construction of the matrix-valued RBF with \({\mathbf {I}}\) being the 3x3 identity matrix, \(\Delta \) being the Laplace operator \(\Delta {=}\nabla ^{2}{=}\nabla {\cdot }\nabla \) and \(\nabla \) is the del operator being \(\nabla {=}[\partial _{x},\partial _{y},\partial _{z}]^{\text {T}}\) in 3D Cartesian coordinates. A scalar RBF \(\phi \) with sufficiently high order can be modified by the differential operator \(\{-\Delta {\mathbf {I}}{+}\nabla \nabla ^{T} \}\) (see (Lowitzsch 2002)) to form the matrix-valued RBF \({\varvec{\Psi } {=} \{-\Delta {\mathbf {I}}{+}\nabla \nabla ^{T} \} \phi }\).In the present study, a Wendland polynomial (Wendland 1995) was chosen for \(\phi \), specifically \(\psi _{4,2}\), which in 1D iswhere \(\psi _{4,2}(r)\) is defined as null for \(r{>}1\). The 3D Cartesian derivatives of \(\psi _{4,2}({\mathbf {r}})\) needed for the construction of \(\varvec{\Psi }\) are as follows:in which the components \(r_{x}\), \(r_{y}\) and \(r_{z}\) are 1D distance components of \({\mathbf {r}}{=}[r_{x},r_{y},r_{z}]^{\text {T}}\) and \(\hat{{\mathbf {r}}}\) is the norm \(\hat{{\mathbf {r}}}{=}\Vert {\mathbf {r}}\Vert \), i.e., the 3D Euclidean distance. The same linear system is set up as in the scalar case to find the coefficient vector \(\lambda \), except now in block-structured formwhereby A now has dimensions 3N\({\times }\)3N, and \(\lambda \) and f have dimensions 3N\(\times \)1. Each block of A, i.e., \(A_{ij}\) is the corresponding \(\Psi _{ij}\) evaluation of \({\mathbf {r}}\) with dimensions N\(\times \)N. The blocks of the vector f are the stacked vector field components, i.e., \(f{=}[u_{1}\ldots u_{N}|v_{1}\ldots v_{N}|w_{1}\ldots w_{N}]^{\text {T}}\) in the present study.

The evaluation of the interpolant \(B \lambda {=} B A^{-1} f\) is analogous to the scalar case except now the block-structured B has dimensions 3M\(\times \)3N and each M\(\times \)N block \(B_{ij}\) is the corresponding \(\Psi _{ij}\) evaluation of \({\mathbf {r}}\), where \({\mathbf {r}}\) is the vector of 1D distances between the known and new points. When the interpolant \(B \lambda \) is evaluated, the resulting vector field (formed by reshaping f from 3M\(\times \)1 to M\(\times \)3) is divergence-free. Formal mathematical derivation of this feature is shown in Lowitzsch (2002) and additional discussion can be found in McNally (2011) and Yang et al. (2014). The divergence-free feature of the output vector field becomes conceptually clear when considering that [u, v, w] at every output point is realized as the linear dot-product sum of weighted basis functions. Since the basis function kernel is constructed as the curl of the curl of a single polynomial function, the divergence of their resulting collocated sum is automatically null by identity \(\nabla {\cdot }(\nabla {\times } \square )= 0\).

The divergence-free RBF interpolation method has been implemented and its increased accuracy with respect to the scalar formulation has been confirmed (Appelbaum 2020). The interpolation of experimental velocity field to a grid block is the first step in the preparation of the unsteady inlet for the DNS.

Following the spatial interpolation of the tracer data to the DNS inlet grid, a series of additional pre-processing steps are necessary to make the data suitable for numerical simulation. These include treatment at the wall boundary as well as periodic boundaries, the solution of the pressure and temperature fields, and finally the non-dimensionalization and subsequent Mach number scaling of the input data.

A major challenge is posed by the finite tracer density near the wall boundary (\(y\,=\,0\)) when the dimensionless wall-distance is considered. The dimensional tracer density is relatively constant, with a slight reduction near the wall due to the tracer bubbles bursting (see Fig. 4), however when considering the local tracer density relative to velocity gradient \(\nabla {\mathbf {u}}\) (or, similarly, spatial tracer density normalized by the local dimensionless wall distance) one sees that near the wall there relatively sparse tracer coverage. For reference, the average nearest neighbor distance \({\bar{r}}_{\text {NN}}{\approx }5.3\text {[mm]}\) (see Sect. 3.2), is equivalent to a wall distance \(y^{+}{\approx }100\), well into the logarithmic layer. This means that, if reduced to a 1D problem, there is on average one single tracer available to resolve the curvature of the velocity profile, including the no-slip condition, below \(y^{+}{\approx }100\) during the interpolation process, which must produce output for every grid point at every timestep. The no-slip condition is therefore not universally respected during the interpolation step (see Sect. 4.1), so measures must be taken to force \({\mathbf {u}}_{y{=}0}{\equiv }{\mathbf {0}}\) to avoid possible discontinuities in the computational domain. For example, one can introduce ‘extra’ artificial tracers having null velocity at the wall grid points for every timestep during interpolation, however this can have the opposite effect in which the null wall velocity becomes over-represented spatially if no real tracers are momentarily in the vicinity to otherwise influence the result, meaning that the velocity profile approaches zero too sharply too far away from the wall. Instead, a wall blending scheme is applied in a post-processing step to force the velocity of the experimental domain to \({\mathbf {u}}_{y{=}0}{\equiv }{\mathbf {0}}\) at the wall. Figure 5 schematically demonstrates this procedure and shows the true proportion between \({\bar{r}}_{\text {NN}}\) and \({\bar{\delta }}_{99}\) in the experimental data. A third-order natural spline (blue) is lofted and blended via a window function to u(y) (red), replacing the lower portion of the velocity profile (red dotted). The black line represents a reference average velocity profile \({\bar{u}}(y)\) at the experimental \(Re_{\theta }\). This procedure is performed for all velocity components u,v and w. A blending function is also used to transition the unsteady field to the steady mean flow field at the span-wise boundaries of the inlet to enforce the periodic condition. Such boundary condition enforcement schemes introduce a certain amount of non-physicality to the incoming flow, the effects of which are discussed in Sect. 6.

In a final step, the flow field variables are non-dimensionalized according to the convention required by NS3D and split up into separate MPI domains for use in the simulation. The dimensionless problem is defined by the characteristic dimensionless quantitieswhere \(Re_{{\bar{\delta }}_{99}}\) is the hydrodynamic boundary layer thickness Reynolds number, Pr is the Prandtl number, \(M_{\infty }\) is the freestream Mach number, \(k_{\infty }\) is the freestream thermal conductivity, \(\rho _{\infty }\) is the freestream density, \(\gamma {=}\frac{c_{p}}{c_{v}}\) is the specific heat ratio and \(c_{\infty }\) is the freestream speed of sound. The freestream velocity magnitude is signified by \(U_{\infty }\) and generally U indicates the velocity magnitude, i.e.,

\(U{=}\Vert {\mathbf {u}}\Vert {=}\hat{{\mathbf {u}}}{=}(u^{2}{+}v^{2}{+}w^{2})^{\frac{1}{2}}\). The freestream dynamic viscosity \(\mu _{\infty }\) is determined by Sutherland’s Law (Sutherland 1893). The length, velocity and time scales are non-dimensionalized bywhere \({\bar{\delta }}_{99}\) is the average inlet hydrodynamic boundary layer thickness. The 3D, unsteady dimensionless temperature, pressure and density fields are calculated as deviations with respect to 1D, dimensionless averaged wall-normal y profiles, which are calculated as follows. The averaged 1D dimensional temperature profile \({\bar{T}}(y)\) is calculated using the Crocco–Busemann relation (Crocco 1932; Busemann 1935; White and Corfield 2006)where \(T_{aw}\) is the dimensional adiabatic wall temperatureand r is the recovery factor \(r{=}Pr^{\frac{1}{3}}\) (Stalder et al. 1950; Ackermann 1942; White and Corfield 2006) with \(Pr{=}0.71\) for air. The low Mach number of the experiment (\(M_{\infty ,{\text {exp}}}= 0.019 \)) means that stagnation heating and compressibility effects in the measured flow field are negligible; therefore, the necessity of deriving the unsteady temperature and density fields is purely numerical and a product of the Mach scaling applied to the dimensionless simulation, as described in Sect. 5.3. With an assumed constant pressure \({\bar{p}}\) over the boundary layer, the density profile is then derived using the equation of state. The inputs to the equations above are from the full (interpolated) transient field which is averaged in the stream-wise (x) and span-wise (z) directions as well as in time so that only a 1D wall-normal profile remains. The 3D unsteady dimensionless temperature, pressure and density fields are then calculated by their unsteady dimensionless, mean-removed \((\square ^{\prime })\) components aswhere the 3D primitive variable field is at all points compared to the corresponding wall-normal (y) position of the averaged profile, indicated by the overlined variables \(({\bar{\square }})\). More details regarding the pre-processing toolchain can be found in Appelbaum (2020).

The flow solver NS3D is used to simulate a ZPGTBL. NS3D is an in-house DNS code developed at the Institute of Aerodynamics and Gas Dynamics (IAG) at the University of Stuttgart. Publications by Babucke (Babucke et al. 2006; Babucke 2009), Linn & Kloker (Linn and Kloker 2008, 2011) and Keller & Kloker (Keller and Kloker 2015, 2017) provide a formal numerical description of the solver. NS3D directly solves the 3D, unsteady, compressible Navier–Stokes equations on a structured rectilinear grid. NS3D uses \(6^{th}\) order compact finite difference (CFTD) operators or \(8^{th}\) order explicit finite differences (Keller and Kloker 2013; Babucke 2009; Dörr 2018) to evaluate discrete spatial derivatives. A \(10^{th}\) order low-pass filter (Visbal and Gaitonde 2002) is used for additional numerical stability. Time integration of the discretized transport equations is performed via the explicit \(4^{th}\) order ‘classical’ Runge–Kutta scheme (Runge 1895; Kutta 1901).

The computational grid is composed of \([576 {\times } 536 {\times } 1280]\) nodes (395M total) in a domain of dimensions \({\bar{\delta }}_{99} \cdot [8.8 {\times } 3.0 {\times } 3.0]\). The mesh and extents of the computational domain were configured in a manner consistent with studies performed in Wenzel et al. (2018); Wenzel (2019). The bottom boundary of the DNS domain is designated as an adiabatic, no-slip wall such that \(({\partial T}/{\partial y})_{y=0} {\equiv } 0\), \({\mathbf {u}}_{y=0} {\equiv } {\mathbf {0}}\) and \(({\partial p}/{\partial y})_{y=0} {\equiv } 0\). The downstream boundary is defined as a subsonic outflow, which requires a steady flow field \(({\partial {\mathbf {u}}}/{\partial t})_{N} {=} ({\partial {\mathbf {u}}}/{\partial t})_{N-1} {\equiv } 0\) and thus the absence of any stream-wise velocity gradient \(({\partial {\mathbf {u}}}/{\partial x}) {\equiv } 0\), assured through the presence of sponge zones. Sponge zones numerically limit the flow field’s deviation from a reference field. The time derivative of the vector of conservative numerical fluxes \({\varvec{Q}}{=}\left[ \rho , \rho u, \rho v, \rho w, E\right] ^{\mathrm {T}}\) is adapted by the relationwhere \(G({\mathbf {x}})\) is a gain parameter. Fluctuations are attenuated by means of an additional source term in the discretized Navier–Stokes (NS) equations which is proportional to the amplitude of the fluctuation of each element of \({\varvec{Q}}\). The gain parameter is a constant multiplicative factor on the attenuative body force. The reference field \({\varvec{Q}}_{\text {ref}}\) is typically the so-called ‘baseflow’ or steady 1D y-normal mean profile (projected in [x, z]) which serves as an initial condition and steady boundary condition where required. The steady baseflow profile is calculated from the averaged experimental flow field, as described in Sect. 4.3. In the present case, an additional unsteady disturbance field is present in the inlet region (the green box in Fig. 3), serving locally as \({\varvec{Q}}_{\text {ref}}\). A unity gain parameter \(G{=}1\) provides a relatively ‘soft’ damping of disturbances with respect to the reference field and is therefore used at the downstream, top and lateral boundaries. A null gain parameter \(G{=}0\) means that no artificial damping of the simulated field is present and is therefore used in the bulk of the simulation domain. Within the inlet region, the reference field \({\varvec{Q}}_{\text {ref}}\) is the unsteady field derived from the 4D-PTV data, as described in Sect. 4.3. Here, a relatively high gain parameter \(0{<}G{<}50\) is used to enforce the externally derived flow field by strongly damping the solution vector’s deviation from the reference field. Figure 6 illustrates the distribution of the sponge gain G in the DNS domain. By \(x^{*}{\approx }0.16\) downstream of the inlet, the sponge gain is null and the DNS solution is thus no longer (directly) forced by the inlet field.

The lateral (\(-z\),\(+z\)) boundaries are defined as periodic. The top boundary is a characteristic freestream boundary, which minimizes the reflection of acoustic waves. Oblique waves are minimized through the transitional grading of G in the boundary-normal direction, as well as through grid stretching in which the growth rate of the grid is increased in a given dimension. A more detailed description of the boundary conditions used in the present case may be found in (Babucke 2009).

The domain is spatially discretized with a rectilinear grid, defined through the projection of three orthogonal 1D coordinate vector arrays. In the present case, the grid is uniformly spaced in the span-wise (z) dimension. In the wall-normal (y) direction, the grid is stretched in the direction of the top boundary (\(+y\)) from a minimal cell size \(\Delta y{=}y_{i+1}{-}y_{i}\) adjacent to the bottom wall. The grid growth rate \(\Delta y_{i+1}{/}\Delta y_{i}\) in the first block of 136 cells near the wall is limited to 1.016 (1.6%), after which it is decreased to 1.00167 (0.167%) until about \(y{/}\delta _{99}{\approx }1.5\), where it is allowed to grow at a rate of 1.008 (0.8%) up to the top boundary. In the stream-wise (x) direction, the grid has a uniform spacing until \(x^{*}{=}x/{\bar{\delta }}_{99}{=}3\), after which it is stretched in the direction of the downstream boundary, promoting numerical damping of turbulent structures as they approach the subsonic outflow boundary. Starred coordinate variables \((\square ^{*})\) indicate the spatial coordinate non-dimensionalized by the average hydrodynamic boundary layer thickness \({\bar{\delta }}_{99}\) of the experimental data at the inlet plane location and ‘plus notation’ \((\square ^{+})\) specifies that a length has been non-dimensionalized by the local viscous length scale \(\nu _{w}{/}u_{\tau }\). The grid resolution has been confirmed to be sufficient through post-processing of the DNS results, which has revealed that the first cell height \(\Delta y_{0}\) is sufficiently small such that \(\Delta y_{0}^{+}{=}(\Delta y_{0} u_{\tau }){/}\nu _{w}\) stays less than roughly 0.60 throughout the complete area of interest (\(x^{*}{<}3\)). In the stream-wise direction, the maximum \(\Delta x^{+}\) is 12 and the maximum \(\Delta z^{+}\) is 3.6.

The sampling period of the experiment (0.001[s]) is roughly 413 times larger than the relatively small timestep size required for the explicit time integration scheme of NS3D. Accordingly, a temporal interpolation scheme is needed to provide data at every sub-timestep of the Runge–Kutta temporal integration scheme, for which a third-order Lagrange polynomial based interpolation is used.

The dimensional and non-dimensional parameters of the experiment and DNS are listed in Table 1. The defined inlet hydrodynamic boundary layer thickness Reynolds number \(Re_{{\bar{\delta }}_{99}}{\approx }43\text {e}3\) is kept constant, however the freestream Mach number \(M_{\infty }\) of the dimensionless problem is increased to \(M_{\infty }{=}0.85\) to reduce the simulation cost. Compared to an equivalent simulation at \(M_{\infty }{=}0.3\), the computational effort at \(M_{\infty }{=}0.85\) is reduced by a factor of roughly three due to the less strict stability constraints on the timestep \(\Delta t_{\text {max}}\) related to the discretization of the energy equation, see Babucke (2009). Error introduced by compressibility effects at \(M_{\infty }{=}0.85\) is predicted from similar studies to not be particularly significant (see Wenzel et al. 2018), and in the present case is likely to fall beneath the magnitude of error introduced during grid interpolation or boundary blending. The dimensionless simulation result is re-dimensionalized using the experimental \(M_{\infty }\) for dimensional comparisons. Quantification of the impact of \(M_{\infty }\) scaling remains to be evaluated in further studies, however the already high agreement between the experimental and DNS flow fields presented in Sect. 7 despite such potential error is encouraging for future studies with lower \(M_{\infty ,\text {DNS}}{/}M_{\infty ,\text {exp}}\).

The flow field at the DNS inlet is reconstructed from 4D-PTV data and lacks some unsteady high-wavenumber content of the laboratory flow. As such, the reconstructed inlet flow field can be analogously described as a spectrally low-pass filtered version of a turbulent flow, as will be demonstrated by filtering DNS results in a post-processing exercise in Sect. 7.2. As the coarsely resolved TBL flow at the domain inlet is introduced to the DNS domain, it undergoes an adjustment process before a fully developed turbulent kinetic energy k spectrum is once again reached (see Fig. 8). As previously mentioned, there are two main drivers for the adjustment effects seen immediately downstream of the inlet: the limited tracer density of the experiment, leading to a lack of high frequency content in the unsteady inflow, and the side effects of boundary blending at the wall.

Non-physical flow structures that don’t fulfill the discretized transport equations are quickly numerically dissipated, as evidenced in Fig. 8b by the immediate reduction in the root-mean-square (RMS) dimensionless wall shear stress \(\tau ^{+}_{w,\text {rms}}\), as well as in (a), where a relatively thick layer of low dimensionless turbulent kinetic energy \(k^{+}\) is present near the wall. The dissipation of such structures and relaxation of shear stresses produces a relaminarizing tendency, as seen in reduction of the skin friction coefficient \(c_{f}{=}2 {\bar{\tau }}_{w}/(\rho _{\infty } U_{\infty }^{2})\) whose minimum occurs at \(x^{*}{\approx }0.7\), downstream of the minimum in \(\tau ^{+}_{w,\text {rms}}\) at \(x^{*}{\approx }0.3\). Due to its non-dimensional representation in wall units, the relaminarizing tendency causes \(k^{+}\) to increased, having a peak ‘bubble’ at \((x^{*}{,}y^{*}){\approx }(1.0{,}0.02)\). Subsequently, the off-wall fluctuations mix back into the near wall region to produce a peak in \(\tau ^{+}_{w,\text {rms}}\) around \(x^{*}{\approx }1.2\). By \(2.5{\lesssim }x^{*}{\lesssim }2.0\), the transient process begins to reach a more or less converged state. Interestingly, the normalized hydrodynamic boundary layer thickness \(\delta _{99}/\delta _{99,\text {exp}}\) grows relatively constantly throughout the domain (Fig. 8b) in spite of this transient process, indicating that the re-adjustment of non-physical conditions at the wall has little effect on the BL thickness.

Such transient effects are also visible in the progression of the various Reynolds numbers in Fig. 8c. The momentum thickness Reynolds number \(Re_{\theta }{=}(U_{\infty } \theta ) / \nu _{\infty }\) initially increases as non-physical structures dissipate, then reduces slightly during the pseudo-relaminarization phase until \(x^{*}{\approx }0.7\), consistent with the minimum in \(c_{f}\) in (b). After \(x^{*}{\gtrsim }1.0\), where turbulent energy production begins to recover, \(Re_{\theta }\) once again increases as expected for a TBL. The progression skin-friction Reynolds number \(Re_{\tau }{=}(u_{\tau } \delta _{99}) / \nu _{w}\) closely resembles that of the skin-friction coefficient \(c_f\) in (b), seeing as both the friction velocity \(u_{\tau }{=}\sqrt{\tau _w / \rho _{w}}\) and \(c_{f}\) are proportional to the wall shear stress \(\tau _{w}{=}\mu (\partial u / \partial y)_{y=0}\). The shape factor \(H_{12}{=}\delta ^{*} / \theta \) relates the displacement thickness \(\delta ^{*}\) and the momentum thickness \(\theta \) and indicates the state of turbulent development of a ZPGTBL. Laminar BLs have higher shape factors, meaning that more deficit in u(y) is present with respect to the freestream \(U_{\infty }\) over a larger range of the boundary layer thickness, and accordingly have less severe wall velocity gradients \((\partial u / \partial y)_{y=0}\). The slight increase in \(H_{12}\) in the range \(0.3{\lesssim }x^{*}{\lesssim }0.7\) is therefore consistent with the previously mentioned relaminarizing tendency.

The turbulent character of the DNS flow field can be efficiently summarized in a composite sense by inspecting the flow in the frequency domain. In the present case, spectral analysis will help to answer a primary question of interest, namely how quickly the recovery of unresolved turbulent scales occurs. Figure 9 presents a wavenumber power spectral density (PSD) of the \(u^{\prime }u^{\prime }\) component of the Re-stress tensor, where \(k_{\ell ,99}\) is the normalized linear wavenumber \(k_{\ell ,99}{=}k_{{\bar{\delta }}_{99}}{/}(2 \pi )={\bar{\delta }}_{99}{/}\lambda \). The 1D span-wise wavenumber spectral analysis is performed on thin volumes at seven stream-wise x locations, each spaced \(0.5{\cdot }{\bar{\delta }}_{99}\) apart. Figure 9a presents the PSD for every available wall-normal position vs normalized wavenumber. As the flow propagates through the DNS domain, the fine scales which are absent at the inlet begin to re-develop, as made visible by the extension of the high power region into the high wavenumber (right) part of the plot with increasing x. Eventually, between \(2.0{\lesssim }x^{*}{\lesssim }2.5\) the shape of the PSD in the \(y^{+}\) vs \(k_{\ell }\) space converges to a final form, indicating that the flow has reached a developed turbulent state.

Figure 9b provides a more quantitative representation of this convergent effect. Here, the spectrum at every selected x-position is plotted in the PSD vs \(k_{\ell }\) space for a number of \(y^{+}\) locations, increasing in distance from the wall. At all chosen \(y^{+}\) locations, the spectra show a convergent trend in the stream-wise direction. It’s also clear that the net gap in energy density is not equal at all wall distances, quantified by the size of the curve envelope. Since the characteristic turbulent structure size is finer approaching the wall, a higher proportion of the spectral energy is concentrated in the high wavenumber regime closer to the wall. It therefore follows that spectral energy gap is wider in this region, as information about high wavenumber modes is limited relative to low wavenumber modes during the construction of the inlet. However, since the finer structures closer to the wall also develop over a shorter characteristic period, the larger spectral gap is closed more quickly, i.e., spectral recovery occurs within a shorter transit distance. Closure of the tighter energy density gap in the outer log layer occurs over a longer distance, as less stress is present to drive the development process. Additionally, the flow speed is higher, so there is less residence time between the fixed stream-wise locations in which the development processes can occur. Nevertheless, by roughly \(2.5{\cdot }{\bar{\delta }}_{99}\) convergent arrival upon a power spectral density profile can be seen for all wall distances.

The instantaneous flow fields of the experiment and DNS can be compared to help qualitatively visualize and support the findings of the spectral analysis in Sect. 6.2. Figure 10 depicts contour plots of the normalized stream-wise velocity \(u{/}U_{\infty }\) on wall-normal slices and wall-parallel slices in the left column, as well as the difference between the DNS and experiment \(\Delta u{/}U_{\infty }\) in the right column. Also present in Fig. 10 is a filtered version of the DNS result, which will be addressed in Sect. 7.2.

Qualitative large scale structural similarity can be clearly seen when the interpolated experimental measurement and DNS domains are compared in Fig. 10, indicating that the goal of reproducing the main characteristics of the instantaneous turbulent field has been met within the considered domain. This result is consistent with the minimal changes seen in the PSD below a cutoff wavenumber of \(k_{\ell ,99}{=}{\bar{\delta }}_{99}/(3{\cdot }{\bar{r}}_{\text {NN}})\) in Fig. 9. The difference in normalized stream-wise velocity \(u / U_{\infty }\) between the DNS result and the interpolated experimental result as seen in the right column in Fig. 10 indicates increasing disparity with increasing stream-wise position x and in the near-wall region. Difference within the inlet region itself is due to the temporal interpolation scheme. The large scale similarity and the ability to identify equivalent structures in both datasets is a highly encouraging result, indicating that at least the large scale, most energetic structures near the TBL edge may be accurately introduced and propagated through a numerical simulation domain of choice. In doing so, an already well developed, realistic boundary condition at the TBL edge is present, fostering the TBL’s quick recovery to a statistically converged state.

A spatial low-pass convolution filter may be applied to the DNS results in an attempt to derive an analogy for the net attenuative characteristic of the experimental procedure and pre-processing toolchain, which are inherently limited in resolution by the tracer density. The aim of such an exercise is to show that the experimental field and its subsequent interpolation to the DNS inlet grid is broadly equivalent to a low-pass filtered DNS result. A simple Gaussian-type filter kernel was chosen, leaving the kernel width \(\sigma \) as a variable parameter. A snapshot of the low-pass filtered DNS result is presented in Fig. 10. The filter operation was run on an x-normal plane at \(x^{*}{=}3.0\), where the DNS solution is appropriately turbulently developed. Figure 12 shows the effect of low-pass filtering on the span-wise wavenumber spectrum, namely that a simple Gaussian ‘blur’ filter with a kernel width \(\sigma /{\bar{r}}_{\text {NN}}=0.4\) applied to a statistically converged DNS result yields a flow that looks qualitatively similar to the interpolated experimental result both in the spatiotemporal (Fig. 10) and wavenumber domains.

Wavenumber domain analysis gives composite information about how the energy density of turbulent fluctuations is distributed across wavenumber modes, however it does not necessarily deliver information about the agreement between the instantaneous flow fields. However, the correlation coefficient between the experimental and DNS flow fields quantifies their normalized covariance and is a metric for their degree of variational similarity in a time-dependent manner. Figure 11 shows the Pearson correlation coefficient r (Pearson 1901) on a span-wise normal slice where for every grid point in [x,y], r is calculated between two time series of velocity magnitude \({\hat{\mathbf {u}}}{=}(u^{2}{+}v^{2}{+}w^{2})^{\frac{1}{2}}\). In Fig. 11a, \({\hat{\mathbf {u}}}(t)\) from the experimental measurement is correlated with that of the filtered DNS result (shown in Fig. 10 and 12) and 11b shows \(r({\hat{\mathbf {u}}})\) between the experiment and the unfiltered DNS result. The time series of the mean-removed, normalized velocity magnitude \({\hat{\mathbf {u}}}^{\prime }(t)/U_{\infty }\) for two points is plotted in (c) and (d). One point is located near the wall where the DNS flow quickly decorrelates from the experimental flow, and the other point is farther from the wall where the flows remain highly correlated. The difference in correlation at the two locations is plainly visible when comparing (c) and (d), though the slightly better correlation of the filtered DNS field is fairly subtle.

In the near wall region, the tracer density is low compared to the local strain. As discussed in Sect. 4.3, this low relative information density necessitates a post-treatment of the interpolated experimental field to ensure that the no-slip condition is restored at the near wall region of the unsteady velocity inlet before simulation. In doing so, artifacts stemming from this treatment are introduced in the region \(y{\lesssim }1{\cdot }{\bar{r}}_{\text {NN}}\) or \(y^{+}{\lesssim }100\), for example an unrealistically steady near-wall profile in which small-scale motions are not resolved. Additionally, the near wall region sees the highest production of turbulent kinetic energy k and the highest frequency of turbulent motion, serving to quickly amplify the deviation from any initially well-correlated state. For these two reasons, one expects that the DNS flow field will quickly become decorrelated with respect to the experimental field in the near-wall region, which is well illustrated in Fig. 11. The decorrelated near-wall flow is transported away from the wall through turbulent diffusive flux. Effectively, a new and statistically independent flow develops at the wall and mixes with the bulk flow. A similar but much weaker effect is seen at the upper boundary of the turbulent inlet, where a blending scheme similar to that used at the wall was used to guarantee a smooth transition to the steady baseflow profile. Accordingly, the artificially ‘blended’ flow at \(y^{*}{>}1.2\) no longer shows correlation to the experimental flow, and is also (albeit more slowly) mixed toward the middle of the BL. As seen in Fig. 10, the large scale turbulent structures in the mid and outer layer are largely preserved throughout the transit length of \(\Delta x^{*}{=}3\) due to their higher momentum and the lower turbulent stress present to drive turbulent mixing, which like the near-wall flow would amplify the degree of decorrelation.

To very roughly estimate the stream-wise distance at which the influence of the decorrelated wall and upper boundary flows permeate the entire simulated TBL, a least squares trend line was calculated using segments from the upper and lower arbitrarily chosen iso-contour of the correlation coefficient \(r{=}0.75\). The iso-contour is plotted in light gray in Fig. 11, the least-squares linear trend line is plotted as a dotted black line and the segment of the \(r{=}0.75\) contour used to generate the trend line is indicated in dark gray. When extrapolated, the linearized contour segments intersect by roughly \(x^{*}{\approx }13.1\), indicating that by this transit distance, one would expect the simulated TBL flow to be largely decorrelated from the experimental flow.