Computational Scaling Architecture & Numerical Validation

GPU-accelerated lattice Boltzmann simulation architecture, Reynolds-number scaling studies, and physics-based numerical validation framework.

Computational Hardware Configuration

The computational platform is designed for high-resolution lattice-based fluid simulation using distributed GPU acceleration.

The primary development configuration consists of:

8× V100

GPU Accelerators

256 GB

Aggregate HBM2 Memory

5.0 GLUPS

Target Throughput

FP64

Precision Mode

The primary performance metric is lattice updates per second rather than traditional mesh-cell throughput.

Lattice Boltzmann Solver Formulation

The solver evolves discrete distribution functions over a velocity lattice.

The lattice update equation is:

\begin{equation*} f_i(\mathbf{x}+\mathbf{c}_i\Delta t,t+\Delta t) = f_i(\mathbf{x},t) - \frac{1}{\tau} (f_i-f_i^{eq}) \end{equation*}

Macroscopic density:

\begin{equation*} \rho=\sum_i f_i \end{equation*}

Velocity:

\begin{equation*} \rho\mathbf{u} = \sum_i f_i\mathbf{c}_i \end{equation*}

The viscosity relationship is:

\begin{equation*} \nu = c_s^2 \left(\tau-\frac12\right) \Delta t \end{equation*}

The architecture supports D3Q19 and D3Q27 lattice models.

Reynolds Number Scaling Envelope

The achievable Reynolds number depends on lattice resolution, relaxation time, collision model, and physical scaling assumptions.

The Reynolds number is:

\begin{equation*} Re=\frac{UL}{\nu} \end{equation*}

DNS MFU Scaling on 8x V100 32GB (Rough Surfaces)
Friction Re (Re_tau)	Grid (Nx x Ny x Nz)	Total Points	Memory Req.	Run Time (8x V100)	Feasibility (256 GB Total VRAM)
Re_tau ~ 180 (Low)	256 x 192 x 128	~ 6.3 Million	~ 6.5 GB	~ 2 - 4 Hours	Safe; ultra-low memory usage
Re_tau ~ 395 (Mod)	512 x 384 x 256	~ 50 Million	~ 51 GB	~ 12 - 24 Hours	Safe; fits easily (~6.4 GB / GPU)
Re_tau ~ 590 (Std)	1024 x 512 x 512	~ 268 Million	~ 274 GB	~ 3 - 5 Days	Borderline; requires precision tuning or unified memory
Re_tau ~ 1000 (High)	2048 x 1024 x 1024	~ 2.1 Billion	~ 2.1 TB	Weeks (Infeasible)	Impossible; drastically exceeds system VRA

These values represent computational scaling estimates. Final results are validated through benchmark problems, conservation tests, and measured GPU performance.

Multi-GPU Domain Decomposition

The solver distributes lattice domains across multiple accelerators.

Configuration:

Accelerator Count: 8 × NVIDIA V100
Aggregate Memory: 256 GB HBM2
Precision: FP64
Parallel Strategy: Domain Decomposition
Scaling Metric: GLUPS

The objective is consistent numerical behavior from single GPU development cases through multi-GPU production simulations.

DNS and Turbulence Validation

Validation focuses on canonical turbulent flow problems where numerical behavior can be compared against established analytical, experimental, and high-fidelity numerical reference solutions.

The validation framework examines several levels of physical behavior:

conservation of mass and momentum
laminar benchmark solutions
turbulent energy evolution
Reynolds stress statistics
turbulent energy spectra
wall-bounded turbulence
separated-flow behavior
wake statistics
viscous dissipation
multiscale flow structure
aerodynamic force prediction
multi-GPU reproducibility

The objective is not simply to reproduce an integrated drag coefficient. The objective is to determine whether the numerical representation preserves the physical structures responsible for the observed aerodynamic behavior.

Conservation Laws

The starting point for validation is conservation of mass:

\begin{equation*} \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i} \left(\rho u_i\right) = 0 \end{equation*}

Conservation of momentum is expressed as:

\begin{equation*} \frac{\partial(\rho u_i)}{\partial t} + \frac{\partial(\rho u_i u_j)}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} \end{equation*}

For a Newtonian fluid, the viscous stress tensor is:

\begin{equation*} \tau_{ij}=2\mu S_{ij} \end{equation*}

where the strain-rate tensor is:

\begin{equation*} S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \end{equation*}

LBM Macroscopic Recovery

The Lattice Boltzmann Method evolves discrete distribution functions rather than directly advancing the macroscopic Navier-Stokes variables.

The density is recovered from the zeroth velocity moment:

\begin{equation*} \rho=\sum_i f_i \end{equation*}

The momentum is recovered from the first moment:

\begin{equation*} \rho u_\alpha = \sum_i f_i c_{i\alpha} \end{equation*}

The equilibrium distribution is:

\begin{equation*} f_i^{eq} = w_i\rho \left[ 1 + \frac{\mathbf{c}_i\cdot\mathbf{u}}{c_s^2} + \frac{(\mathbf{c}_i\cdot\mathbf{u})^2}{2c_s^4} - \frac{\mathbf{u}\cdot\mathbf{u}}{2c_s^2} \right] \end{equation*}

For the single-relaxation-time formulation:

\begin{equation*} f_i(\mathbf{x}+\mathbf{c}_i\Delta t,t+\Delta t) = f_i(\mathbf{x},t) - \frac{1}{\tau} \left[ f_i(\mathbf{x},t)-f_i^{eq}(\mathbf{x},t) \right] \end{equation*}

The kinematic viscosity is related to the relaxation time by:

\begin{equation*} \nu = c_s^2 \left( \tau-\frac{1}{2} \right)\Delta t \end{equation*}

Turbulent Decomposition

For turbulent flow, the instantaneous velocity can be decomposed into mean and fluctuating components:

\begin{equation*} u_i=\overline{u_i}+u_i' \end{equation*}

The nonlinear velocity product becomes:

\begin{equation*} \overline{u_i u_j} = \overline{u_i}\,\overline{u_j} + \overline{u_i'u_j'} \end{equation*}

The Reynolds stress tensor is related to the velocity fluctuations:

\begin{equation*} R_{ij} = \overline{u_i'u_j'} \end{equation*}

The turbulent kinetic energy is:

\begin{equation*} k = \frac{1}{2} \overline{u_i'u_i'} \end{equation*}

or, in Cartesian coordinates:

\begin{equation*} k = \frac{1}{2} \left( \overline{u'^2} + \overline{v'^2} + \overline{w'^2} \right) \end{equation*}

Strain and Vorticity

The velocity-gradient tensor can be decomposed into symmetric and antisymmetric components:

\begin{equation*} \frac{\partial u_i}{\partial x_j} = S_{ij}+\Omega_{ij} \end{equation*}

where:

\begin{equation*} S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \end{equation*}

and:

\begin{equation*} \Omega_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \end{equation*}

The vorticity is:

\begin{equation*} \omega_i = \epsilon_{ijk} \frac{\partial u_k}{\partial x_j} \end{equation*}

The strain field identifies regions where velocity gradients produce viscous deformation, while the rotation tensor describes the local rotational component of the flow.

Energy Dissipation

The local viscous dissipation rate per unit mass is:

\begin{equation*} \epsilon = 2\nu S_{ij}S_{ij} \end{equation*}

The corresponding dissipation per unit volume is:

\begin{equation*} \Phi = 2\mu S_{ij}S_{ij} \end{equation*}

The dissipation field provides a spatially resolved measure of where kinetic energy is converted into internal energy.

This makes dissipation particularly useful for multiscale analysis. Regions of elevated dissipation can be compared with:

boundary-layer structures
turbulent shear layers
separated regions
coherent vortical structures
wake structures
surface roughness
regions of aerodynamic loss

Lighthill Acoustic Analogy

The viscous dissipation rate should not be confused with the Lighthill stress tensor.

The dissipation rate is a scalar quantity describing viscous conversion of kinetic energy:

\begin{equation*} \epsilon = 2\nu S_{ij}S_{ij} \end{equation*}

Lighthill's acoustic analogy instead describes aerodynamic sound generation through an acoustic wave equation:

\begin{equation*} \frac{\partial^2\rho'}{\partial t^2} - c_0^2\nabla^2\rho' = \frac{\partial^2 T_{ij}} {\partial x_i\partial x_j} \end{equation*}

where the Lighthill stress tensor is:

\begin{equation*} T_{ij} = \rho u_i u_j - \tau_{ij} + \delta_{ij} \left[ p-p_0 - c_0^2(\rho-\rho_0) \right] \end{equation*}

The Lighthill tensor therefore contains contributions associated with nonlinear momentum transport, viscous stresses, and thermodynamic fluctuations.

The distinction is important for the Base Drag framework.

The dissipation field describes where turbulent kinetic energy is being transferred and dissipated.

The Lighthill tensor describes an effective acoustic source distribution.

They are different physical quantities, although both can be evaluated from the same underlying flow solution.

Turbulent Energy Spectrum

The velocity field can be transformed into Fourier space:

\begin{equation*} \widehat{\mathbf{u}}(\mathbf{k}) = \mathcal{F} \left[ \mathbf{u}(\mathbf{x}) \right] \end{equation*}

The kinetic-energy spectrum can then be represented as:

\begin{equation*} E(k) = \frac{1}{2} \sum_{|\mathbf{k}|=k} \left| \widehat{\mathbf{u}}(\mathbf{k}) \right|^2 \end{equation*}

The total turbulent kinetic energy is related to the spectrum through:

\begin{equation*} K = \int_0^\infty E(k)\,dk \end{equation*}

For homogeneous isotropic turbulence, the inertial-range scaling is commonly represented by:

\begin{equation*} E(k) \propto \epsilon^{2/3}k^{-5/3} \end{equation*}

The spectrum is treated as a diagnostic of energy distribution across scales rather than as a requirement that every aerodynamic flow produce an ideal inertial range.

Wavelet Analysis

Fourier analysis describes the distribution of energy across wavenumbers but provides limited spatial localization.

Wavelet analysis provides a complementary representation in which scale and spatial location are retained simultaneously.

A field such as turbulent dissipation can be represented using a wavelet basis:

\begin{equation*} \epsilon(\mathbf{x}) = \sum_{j,k} a_{j,k}\psi_{j,k}(\mathbf{x}) \end{equation*}

where j represents scale and k represents spatial location.

The wavelet coefficient is:

\begin{equation*} a_{j,k} = \int \epsilon(\mathbf{x}) \psi_{j,k}(\mathbf{x}) \,d\mathbf{x} \end{equation*}

For a dyadic wavelet decomposition, the characteristic length scale can be written as:

\begin{equation*} \ell_j = \ell_0 2^{-j} \end{equation*}

Increasing j therefore corresponds to progressively smaller spatial structures.

The result is a hierarchical representation of the turbulent field.

Wavelet Energy

The energy contained at a particular wavelet scale can be estimated from the wavelet coefficients:

\begin{equation*} E_j = \sum_k \left| a_{j,k} \right|^2 \end{equation*}

The resulting sequence

\begin{equation*} E_0,E_1,E_2,\ldots,E_J \end{equation*}

provides a compact representation of how the field is organized across spatial scales.

Unlike a purely Fourier representation, the wavelet coefficients retain information about where structures occur in physical space.

Multiscale Correlation

The wavelet representation also permits correlations between different physical scales to be examined.

A scale-to-scale correlation can be written as:

\begin{equation*} C(j,j') = \left\langle a_{j,k}a_{j',k'} \right\rangle \end{equation*}

A normalized correlation coefficient can be defined as:

\begin{equation*} \rho(j,j') = \frac{ \left\langle (a_j-\overline{a_j}) (a_{j'}-\overline{a_{j'}}) \right\rangle }{ \sigma_j\sigma_{j'} } \end{equation*}

This provides a quantitative method for investigating whether structures at different physical scales remain statistically related.

For the Base Drag research direction, this analysis is important because persistent hierarchical organization could indicate that the effective number of variables controlling an aerodynamic phenomenon is substantially smaller than the number of variables required to represent the complete flow field.

Aerodynamic Forces

The final validation step connects the resolved flow structures to measurable aerodynamic forces.

The surface traction is:

\begin{equation*} t_i = \sigma_{ij}n_j \end{equation*}

where the total stress tensor is:

\begin{equation*} \sigma_{ij} = -p\delta_{ij} + \tau_{ij} \end{equation*}

The total surface force is:

\begin{equation*} F_i = \int_S \sigma_{ij}n_j \,dS \end{equation*}

The drag force is the component of the total force in the flow direction:

\begin{equation*} D = \mathbf{F}\cdot\widehat{\mathbf{e}}_D \end{equation*}

The drag coefficient is:

\begin{equation*} C_D = \frac{D} {\frac{1}{2}\rho U_\infty^2 A} \end{equation*}

This provides the connection between the multiscale flow representation and the engineering quantity being optimized.

Canonical DNS Validation

The computational framework will be evaluated using canonical turbulent flows for which reference data are available.

Validation targets include:

conservation of mass
conservation of momentum
laminar benchmark solutions
turbulent decay
kinetic-energy evolution
Reynolds stresses
velocity statistics
pressure statistics
turbulent energy spectra
dissipation statistics
wall-bounded velocity profiles
skin-friction coefficients
separated-flow statistics
wake statistics
aerodynamic force coefficients
wavelet-scale distributions
multiscale correlations

A simulation that reproduces only the mean velocity field is not considered fully validated.

Likewise, agreement in an integrated drag coefficient does not by itself demonstrate that the underlying turbulent structure has been correctly reproduced.

The validation framework therefore evaluates both integrated engineering quantities and the physical structures from which those quantities emerge.

Multi-GPU Reproducibility

The intended computational architecture includes distributed multi-GPU simulations.

The physical domain can be decomposed into subdomains:

\begin{equation*} \Omega = \bigcup_{m=1}^{N_{GPU}} \Omega_m \end{equation*}

Each GPU advances its local lattice while exchanging information with neighboring subdomains.

The same physical problem can therefore be simulated using different domain decompositions.

Reproducibility can be evaluated through quantities such as:

\begin{equation*} \Delta C_D = \left| C_D^{(N_1)} - C_D^{(N_2)} \right| \end{equation*}

Equivalent comparisons can be performed for velocity statistics, pressure, dissipation, energy spectra, wavelet coefficients, and surface forces.

The objective is physical reproducibility rather than requiring identical floating-point operation ordering between different parallel decompositions.

Validation to Optimization

The validation architecture establishes a progression from numerical correctness to physically meaningful optimization variables.

The intended progression is:

Simulate → Validate → Decompose → Identify Structure → Represent → Optimize

The objective is therefore not simply to produce a high-resolution flow field.

The objective is to determine whether the simulation contains repeatable physical structure that can be extracted, represented, and connected to aerodynamic performance.

If such structure can be identified, it may provide a physics-based means of reducing the effective dimensionality of aerodynamic design optimization.

This establishes the connection between DNS, multiscale analysis, reduced-order modeling, and the broader Base Drag technology development program.

Validation Roadmap

The development sequence is:

Solver Verification
       |
       v
Single GPU Validation
       |
       v
Multi-GPU Scaling
       |
       v
DNS Benchmark Comparison
       |
       v
Surface Interaction Studies
       |
       v
Reduced Order Modeling

This architecture provides a foundation for studying complex fluid systems through high-performance simulation and multiscale analysis.

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