Viscous Terms Research Articles

Potential of data-driven discovery of nonlinear wave equations

Data-driven partial differential equation discovery techniques have been growing in importance in recent years. This paper discusses its possible applications in the field of finite-amplitude sound propagation and the related phenomena (such as generation of higher harmonics and shock formation). Two cases are investigated, namely the propagation of pressure pulses as travelling waves and the interference of pressure pulses, the former leading to the Westervelt equation and the latter to the Kuznetsov equation. It is shown how representative wave equations can be extracted from data obtained by numerically solving the compressible Navier-Stokes equations (up to the third order smallness in entropy changes) and subsequently employing the sparsity promoting regression techniques. The resulting error of the equation coefficients is about 7% in the nonlinear terms. The limitations of this approach are discussed (e.g., the appropriate capturing of thermal and viscous loss terms).

Investigating thermal conduction dynamics in ternary Carreau–Yasuda nanofluids under convective boundary conditions and exponential heat source/sink effects

The analysis of the Carreau–Yasuda nanofluid (CYNF) across a stretching surface has practical applications in several fields, such as heat exchange, engineering, and material science. Scholars may discover novel perspectives for increasing heat transfer performance, establishing advanced materials, and enhancing the efficiency of multiple engineering systems by studying the behavior of CYNF in this particular instance. The energy transfer through trihybrid CYNF flow with the effect of magnetic dipole across a stretching sheet is examined in the present study. The ternary hybrid nanofluid (THNF) has been prepared by the addition of ternary nanoparticles (NPs) in the water (50%) and ethylene glycol (50%). Titanium dioxide (TiO2), silicon dioxide (SiO2), and aluminum oxide (Al2O3) are used in base fluid. The fluid velocity and heat transfer are examined under the impact of Darcy–Forchheimer, chemical reaction, convective condition, activation energy, and exponential heat source. The fluid flow has been stated in form of velocity, energy, and concentration equations. The set of modeled equations is simplified to non-dimensional form of ordinary differential equations (ODEs) by using similarity substitutions. Numerically, the system of lowest order ODEs is calculated through the parametric continuation method. It has been noticed that the temperature field augments against the variation of viscous dissipation and heat source term. Furthermore, the magnetic dipole has significant impact to enhance the thermal curve of THNF whereas falloffs the velocity profile. It can be noticed that the energy propagation rate enhances with the rising numbers of ternary NPs, from 1.22% to 5.98% (nanofluid), 2.30% to 9.61% (hybrid nanofluid), and 3.23% to 11.12% (trihybrid nanofluid).

Adaptive-coefficient finite difference frequency domain method for time fractional diffusive-viscous wave equation arising in geophysics

The diffusive-viscous wave (DVW) equation is a widely used model to describe frequency dependent attenuation of seismic wave in fluid-saturated porous medium. In this paper taking power law frequency dependent attenuation into account, we first introduce a modified DVW equation (time fractional DVW equation) in which the first order temporal derivative of viscous term is replaced with a fractional order temporal derivative. In consideration of that most of the existing numerical simulations for seismic wave equations are based on time domain methods and truncation with some specific boundary conditions, we incorporate the absorbing boundary condition as complex-frequency-shifted (CFS) perfectly matched layer (PML) into the time fractional DVW equation, and then develop an adaptive-coefficient (AC) finite difference frequency domain (FDFD) method for numerical simulation. The corresponding analytical solution for homogeneous time fractional DVW equation is provided for model validation, and the effectiveness of the developed AC FDFD method is verified by some numerical examples including the homogeneous model and the layered model. Numerical results show that AC FDFD method is more accurate than the traditional 2nd-order FDFD method for numerical modeling of time fractional DVW equation with CFS PML absorbing boundary condition, while requiring similar computational costs.

Uniform solitary wave theory for viscous flow over topography

The flow of a density stratified fluid over obstacles has been intensively explored from a natural and scientific point of view. This flow has been successfully governed by using the forced Korteweg–de Vries-Burgers equation that generated solitons in a viscous flow. This is done by adding the viscous term beyond the Korteweg–de Vries approximation. It is based on the conservation laws of the Korteweg–de Vries-Burgers equation for mass and energy, and assumes that the upstream wavetrains are composed of solitary waves. Our results show that the influence of viscosity plays a key role in determining the upstream solitary wave amplitude of the bore. A good comparison is obtained between the numerical and analytical solutions.

An improved detached eddy simulation method for cavitation multiphase flow

The evolution of cavitation flow involves intense transient characteristics and complex multi-scale motions, significantly increasing the difficulty and computational cost of solving in separation zones. This study proposes an Improved Detached Eddy Simulation (IDES) method, based on a single-equation eddy-viscosity model, aimed at enhancing the resolution of small-scale cavitation flows. By incorporating the broad-spectrum von Karman scale concept, the method effectively improves the resolution capability for small-scale flows in separated zones. The performance of the IDES model was validated through three computational cases, and the main conclusions are as follows: The results of the triangular cylinder flows show that the IDES model demonstrates a resolution capability for large-scale turbulence comparable to the Large Eddy Simulation (LES) model. The orifice flow results indicate that activating the LES method with insufficient grid resolution is highly discouraged, whereas the IDES model performs more robustly under such conditions. For hydrofoil flows, the IDES model more accurately captures the temporal evolution of cavitation, avoiding the premature cavity shedding predicted by LES under coarse grid conditions. The decomposition results of the vorticity transport equation show that both the dilatation term and viscous term are significant contributors in the vorticity transport process, with their average contributions to vorticity reaching 49.25% and 39.99%, respectively. Vortices can be considered the primary controllers of cavitation dynamics, while the dilatation term and viscous term can be regarded as the main controllers of vorticity. Overall, the energy compensation mechanism of the IDES model, based on local flow information, gives it unique advantages in handling small-scale turbulence and cavitation phenomena, providing both high computational efficiency and accuracy.

Effects of Externally Applied Stress on Multiphase Flow Characteristics in Naturally Fractured Tight Reservoirs

Externally applied stress on the rock matrix plays a crucial role in oil recovery from naturally fractured tight reservoirs, as local variations in pore pressure and in-situ tension are expected. The published literature severely lacks in evaluations of the characteristics of hydrocarbons, displaced by water, in fractured reservoirs under the action of externally applied stress. This study intends to overcome this knowledge gap by resolving complex time- and stress-dependent multiphase flow by employing a coupled Finite Element Method (FEM) and Computational Fluid Dynamics (CFD) solver. Extensive three-dimensional numerical investigations have been carried out to estimate the effects of externally applied stress on the multiphase flow characteristics at the fracture–matrix interface by adding a viscous loss term to the momentum conservation equations. The well-validated numerical predictions show that as the stress loading increases, the porosity and permeability of the rock matrix and capillary pressure at the fracture–matrix interface decrease. Specifically, matrix porosity decreases by 0.13% and permeability reduces by 1.3% as stress increases 1.5-fold. Additionally, stress loading causes a decrease in fracture permeability by up to 29%. The fracture–matrix interface becomes more water-soaked as the stress loading on the rock matrix increases, and thus, the relative permeability curves shift to the right.

A generalized reduced-order model for trans-stenotic pressure drop with and without a guidewire

Guidewire-based pressure measurement is essential for diagnosing coronary artery disease. However, the impact of the guidewire on local hemodynamics and diagnostic outcomes is not fully understood. In this study, we propose a generalized reduced-order model (ROM) to accurately predict the trans-stenotic pressure drop in arteries. A key advantage of this model is that the viscous term does not rely on empirical parameters, making it applicable to both scenarios with and without guidewire insertion, and across varying stenosis severities. The proposed model demonstrates good accuracy compared to 3D idealized numerical models, achieving an average prediction error of 3.61% for cases without a guidewire and 4.53% for cases with a guidewire. Furthermore, when applied to a patient-specific model, it achieves comparable or better results than previously published ROMs. Finally, this ROM is employed to investigate the shifting relative importance of different components of the trans-stenotic pressure drop at various stenosis severities, and to provide further insights into the guidewire’s influence on FFR measurements.

Heat transfer analysis of nanofluid flow with entropy generation in a corrugated heat exchanger channel partially filled with porous medium

AbstractHeat exchanger research is mainly exploited to develop and optimize new engineering systems with high thermal efficiency. Passive methods based on nanofluids, fins, wavy walls, and the porous medium are the most attractive ways to achieve this goal. This investigation focuses on heat transfer and entropy production in a nanofluid laminar flow inside a plate corrugated channel (PCC). The channel geometry comprises three sections, partially filled with a porous layer located at the intermediate corrugate channel section. The physical modeling is based on the laminar, two‐dimensional Darcy–Brinkman–Forchheimer formulation for nanofluid flow and the local thermal equilibrium model for the heat equation, including the viscous dissipation term. Numerical solutions were obtained using ANSYS Fluent software based on the finite volume technique and the appropriate meshed geometries. The numerical results are validated with theoretical, numerical, and experimental studies. The simulations are performed for CuO–water nanofluid and AISI 304 porous medium. The coupled effects of porous layer thickness (δ), Reynolds number (Re), and nanoparticle fraction (φ) on velocity, streamlines, isotherm contours, Nusselt number (Nu), and entropy generation (S) are analyzed and illustrated. The simulation results demonstrate that heat transfer enhancement in clear PCC can be achieved using a porous layer insert. For the porous thickness range of [0.1–0.6], the corresponding range of average Nusselt number increase is [35.7%–176.9%], and the average entropy generation is [105.4%–771.9%]. The effect of the Reynolds number is more important in a porous duct than in a clear one. For δ = 0.4 and φ = 5%, the increase of Re in the range of [200–500] induces an increase in average Nusselt number in the range of [80.9%–108.4%] and average entropy in [222.9%–309.1%] comparatively to clear PCC. The effect of φ is practically the same for porous and clear channels. For φ = 5%, the increase on average Nu is about 9%, and entropy generation is 5%. Accordingly, important improvements in heat transfer in PCC can be achieved through the combined effect of flow Reynolds number and porous layer thickness.

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.

Local quadratic convergence of the SQP method for an optimal control problem governed by a regularized fracture propagation model

We prove local quadratic convergence of the sequential quadratic programming (SQP) method for an optimal control problem of tracking type governed by one time step of the Euler-Lagrange equation of a time discrete regularized fracture or damage energy minimization problem. This lower-level energy minimization problem % describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process and a viscous regularization term. Conditions on the latter, corresponding to a time step restriction, guarantee strict convexity and thus unique solvability of the Euler Lagrange equations. Nonetheless, these are quasilinear and the control problem is nonconvex. For the convergence proof with $L^\infty$ localization of the SQP-method, we follow the approach from \cite{Troeltzsch:1999}, utilizing strong regularity of generalized equations and arguments from \cite{HoppeNeitzel:2020} for $L^2$-localization. 2020 Mathematics Subject Classification 90C55, 49M41, 49M15.

Existence of Regular Solutions for a Class of Incompressible Non-Newtonian MHD Equations Coupled to the Heat Equation

We consider a system of PDE’s describing the steady flow of an electrically conducting fluid in the presence of a magnetic field. The system of governing equations composes of the stationary non-Newtonian incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy is taken into account in the momentum equation and the Joule heating and viscous heating terms are included. We proved the existence of C1,γ(Ω¯)×W2,r(Ω)×W2,2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }({\\bar{\\Omega }})\ imes W^{2,r}(\\Omega )\ imes W^{2,2}{(\\Omega )}$$\\end{document} solutions of the systems for 1<p<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1< p<2$$\\end{document} corresponding to a small data and we show that this solution is unique in case 6/5<p<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6/5< p < 2$$\\end{document}. Moreover, we also proved the higher regularity properties of this solution.

Modelling internal erosion using 2D smoothed particle hydrodynamics (SPH)

This paper presents a stabilised multi-phase smoothed particle hydrodynamics (SPH) model applicable to seepage-induced internal erosion and the resulting deformation in soils. Based on the continuum mixture theory, a new single-layer SPH model in the u-w-p formulation is derived for the mathematical description of the erodible porous material. A new modified constitutive model is formulated to account for the influence of erosion on the mechanical behaviour of soils. Extensions of a first-order consistent boundary condition as well as the diffusion algorithm for hydro-mechanical coupling in SPH are also proposed to accommodate the analysis of erosion-transport process. A novel viscous dissipation term is designed to mitigate the numerical instability commonly encountered in coupled problems. In contrast to existing stabilisation terms in the SPH literature, which operate on both the bulk component and shear component, this new stabilisation term offers the possibility to increase the dissipation of unphysical oscillation due to the shear deformation. The proposed method is validated through a series of benchmark tests. The results demonstrate the effectiveness of the proposed approach in addressing the coupled problem in porous materials involving multi-phase flow, phase interaction/transfer, and large deformation with enhanced accuracy, stability, and robustness.

Model predictive control of a wave-to-wire wave energy converter system with non-linear dynamics and non-linear constraints using a tailored pseudo-spectral method

A novel pseudo-spectral method is proposed to tackle the non-linear control problem of a wave energy converter (WEC) integrated with an all-electrical power-take-off (PTO). The WEC hydrodynamic model is constructed using linear wave theory together with a non-linear viscous damping term. The PTO model encompasses a permanent magnet synchronous generator (PMSG) whose local field weakening mechanism imposes non-linear operational constraints on the WEC’s motion and PTO torque. The energy-maximizing control problem of this wave-to-wire model is non-linear and has not been investigated before. In comparison to pseudo-spectral methods previously studied, the proposed pseudo-spectral method is exempt from the assumption of ideal incoming wave prediction. This is made possible by the introduction of a novel mapping function to convert the finite control horizon onto the Gaussian quadrature interval. This mapping function increases more slowly with time at the beginning of the control horizon where wave prediction attains greater precision. As a result, the collocation points are utilized to the advantage of the better approximation of the objective function, thus better overall control performance.

Axial Green Function Method in an Efficient Projection Scheme for Incompressible Navier–Stokes Flow in Arbitrary Domains

We introduce a new approach to solving incompressible Navier–Stokes flow. This method combines a projection scheme with the Axial Green Function Method (AGM). Based on the Kim and Moin methods, our methodology employs a predictor–corrector mechanism to achieve stable and accurate velocity corrections. Using axial Green functions, we transform complex differential equations into simpler one-dimensional integral equations. These are strategically placed along a minimal axis-parallel lines, known as axial lines, within the flow domain. This transformation makes computation and analysis more efficient from a numerical viewpoint. A significant innovation in our approach is using one-dimensional axial Green functions tailored explicitly for the reaction–diffusion ordinary differential operator. These functions efficiently handle the discrete-time derivative and viscous terms of the momentum equation. Furthermore, our approach allows for the arbitrary construction of axis-parallel lines, facilitating analysis near critical flow regions and even enabling the random distribution of these lines. We validate our proposed method through numerical examples, demonstrating the convergence of numerical solutions, the effectiveness of arbitrarily constructed axis-parallel lines, and the potential extension of our method to three-dimensional flow problems. Additionally, this study provides a robust and adaptable alternative way to solve incompressible Navier–Stokes flows, proving its effectiveness in practical applications such as Tesla valves.

Intermittency in the not-so-smooth elastic turbulence.

Elastic turbulence is the chaotic fluid motion resulting from elastic instabilities due to the addition of polymers in small concentrations at very small Reynolds ( ) numbers. Our direct numerical simulations show that elastic turbulence, though a low phenomenon, has more in common with classical, Newtonian turbulence than previously thought. In particular, we find power-law spectra for kinetic energy E(k) ~ k-4 and polymeric energy Ep(k) ~ k-3/2, independent of the Deborah (De) number. This is further supported by calculation of scale-by-scale energy budget which shows a balance between the viscous term and the polymeric term in the momentum equation. In real space, as expected, the velocity field is smooth, i.e., the velocity difference across a length scale r, δu ~ r but, crucially, with a non-trivial sub-leading contribution r3/2 which we extract by using the second difference of velocity. The structure functions of second difference of velocity up to order 6 show clear evidence of intermittency/multifractality. We provide additional evidence in support of this intermittent nature by calculating moments of rate of dissipation of kinetic energy averaged over a ball of radius r, εr, from which we compute the multifractal spectrum.

A thermal energy analysis of binary (Go-Co/H2O) and ternary (Go-Co-Zro2/H2O) nanofluids based on characterization and thermal performance

Hybrid or binary nanofluids have superior mechanical and thermal characteristics but the tri-hybrid nanofluids comprise of more embellished thermal properties, better physical strength, and enhanced stability. The present work characterizes the thermal and physical aspects of the hybrid and tri-hybrid nanofluids. The nano-composition of graphene oxide ( Go) and cobalt ( Co) is used in the amalgamation of hybrid nanofluid Go-Co/H2O, whereas the addition of zirconium oxide ( ZrO2) in this mixture gives rise to the ternary Go-Co-ZrO2/H2O hybrid nanofluid. The activation energy and viscous dissipation terms are also amended in the governing equations. The mathematical framework consists of a complex natured dynamical system. However, a numerical algorithm based on finite-difference discretization is developed which can solve the system numerically via the MATLAB software. A comparison with the existing literature is provided in order to validate the numerical procedure. From the outcomes, it is noticed that the temperature of hybrid as well as tri-hybrid nanofuid increases rapidly with change in concentration of zirconium oxide and cobalt. Temperature increases up to 20% by taking 0.1 volume fraction of both zirconium oxide and cobalt. Porous medium and activation energy resist the flow and concentration respectively. A comparative judgment evidently reveals that tri-hybrid Go-Co-ZrO2/H2O nanofluid has a substantial effect on temperature as equated to hybrid or pure nanofluid.

A generalized curvilinear solver for spherical shell Rayleigh-Bénard convection

Summary A three-dimensional finite-difference solver has been developed and implemented for Boussinesq convection in a spherical shell. The solver transforms any complex curvilinear domain into an equivalent Cartesian domain using Jacobi transformation and solves the governing equations in the latter. This feature enables the solver to account for the effects of the non-spherical shape of the convective regions of planets and stars. Apart from parallelization using MPI, implicit treatment of the viscous terms using a pipeline alternating direction implicit scheme and HYPRE multigrid accelerator for pressure correction makes the solver efficient for high-fidelity direct numerical simulations. We have performed simulations of Rayleigh-Bénard convection at two Rayleigh numbers Ra = 105 and 107 while keeping the Prandtl number fixed at unity (Pr = 1). The average radial temperature profile and the Nusselt number match very well, both qualitatively and quantitatively, with the existing literature. Closure of the turbulent kinetic energy budget, apart from the relative magnitude of the grid spacing compared to the local Kolmogorov scales, ensures sufficient spatial resolution.

Viscous correction to the potential flow analysis of Rayleigh–Taylor instability in a Rivlin–Ericksen viscoelastic fluid layer with heat and mass transfer

AbstractThe current investigation focuses on examining viscous corrections for viscous potential flow (VCVPF) analysis concerning the Rayleigh–Taylor instability occurring at the interface of a Rivlin–Ericksen (R–E) viscoelastic fluid and a viscous fluid during the transfer of heat and mass between phases. The R–E model is a fundamental framework in the study of viscoelastic fluids, providing insights into their complex rheological behavior. It characterizes the material's response to both deformation and flow, offering valuable predictions for various industrial and biological applications. Within the framework of viscous potential flow (VPF) theory, viscosity is exclusively accounted for in the normal stress balance equation, disregarding the influence of shearing stress entirely. This study introduces a viscous pressure term into the normal stress balance equation alongside the irrotational pressure, presuming that this addition will improve the discontinuity of tangential stresses at the fluid interface. Through derivation of a dispersion relationship and subsequent theoretical and numerical stability analyses, the stability of the interface is investigated across various physical parameters. Multiple plots are generated using the dispersion relation, and a comparative analysis between VPF and VCVPF is conducted to establish improved stability criteria. The investigation highlights that the combined impact of heat/mass transport and shearing stress serves to delay the instability of the interface.

What Rayleigh numbers are achievable under Oberbeck–Boussinesq conditions?

The validity of the Oberbeck–Boussinesq (OB) approximation in Rayleigh–Bénard (RB) convection is studied using the Gray &amp; Giorgini (Intl J. Heat Mass Transfer, vol. 19, 1976, pp. 545–551) criterion that requires that the residuals, i.e. the terms that distinguish the full governing equations from their OB approximations, are kept below a certain small threshold $\hat {\sigma }$ . This gives constraints on the temperature and pressure variations of the fluid properties (density, absolute viscosity, specific heat at constant pressure $c_p$ , thermal expansion coefficient and thermal conductivity) and on the magnitudes of the pressure work and viscous dissipation terms in the heat equation, which all can be formulated as bounds regarding the maximum temperature difference in the system, $\varDelta$ , and the container height, $L$ . Thus for any given fluid and $\hat {\sigma }$ , one can calculate the OB-validity region (in terms of $\varDelta$ and $L$ ) and also the maximum achievable Rayleigh number ${{Ra}}_{max,\hat {\sigma }}$ , and we did so for fluids water, air, helium and pressurized SF $_6$ at room temperature, and cryogenic helium, for $\hat {\sigma }=5\,\%$ , $10\,\%$ and $20\,\%$ . For the most popular fluids in high- ${{Ra}}$ RB measurements, which are cryogenic helium and pressurized SF $_6$ , we have identified the most critical residual, which is associated with the temperature dependence of $c_p$ . Our direct numerical simulations (DNS) showed, however, that even when the values of $c_p$ can differ almost twice within the convection cell, this feature alone cannot explain a sudden and strong enhancement in the heat transport in the system, compared with its OB analogue.

Integral methods for friction decomposition and their extensions to rough-wall flows

A primary objective of integral methods, such as the momentum integral method, is to discern the physical processes contributing to skin friction. These methods encompass the momentum, kinetic energy and angular momentum integrals. This paper reformulates existing integrals based on the double-averaged Navier–Stokes equations, and extends their application to flows over rough walls. Our derivation yields distinct decompositions for the bottom-wall viscous friction coefficient, denoted as $C_S$ , and the roughness element drag coefficient $C_R$ . The decompositions comprise three terms: a viscous term, a turbulent term and a roughness (dispersive) term – regardless of the flow configuration, be it channel or boundary layer. Notably, when these integrals are evaluated for laminar flow scenarios, only the viscous term remains significant. In addition, we elucidate the spatial distributions of the terms within these decompositions. To demonstrate the practicality of our formulations, we apply them to analyse data from direct numerical simulations of turbulent half-channel flows. These flows feature aligned and staggered cubical roughness at various packing densities. Our analyses, based on kinetic-energy-oriented decompositions, reveal that when the surface coverage density $\lambda _p$ is small, the dominant terms within the decompositions are the viscous and turbulent terms. With increasing $\lambda _p$ , the viscous dissipation term decreases, while the turbulent production term increases and then decreases. These variations arise from a subdued near-wall cycle and the development of a shear layer at the height of the cubes.