A decomposition of available potential energy is derived for Boussinesq fluid flow in networks of connected control volumes. The two constituent parts of the decomposition are positive definite and therefore meaningful representations of available energy. The first (inner) part accounts for available potential energy that is intrinsic to each control volume, while the second (outer) part accounts for the context provided by the larger parent volume to which each smaller control volume belongs. While the intended application casts the control volumes as connected rooms in a building, the formulation can be applied to any domain that is partitioned by either physical boundaries or abstract zones and can be invoked recursively to clarify the hierarchical dependence of available potential energy on scale and context. By deriving budgets for the decomposition, two ways in which available potential energy can be redistributed between its inner and outer parts are identified. The first accounts for an apparent generation of available potential energy due to diapycnal mixing within a control volume that is constrained by removable boundaries. The second involves the reversible conversion between inner and outer parts that occurs when mass or heat is transported between control volumes and accounts for the concomitant change in context. Analytical expressions are derived for the hierarchy of contributions to available potential energy in an example involving three connected spaces, before budgets for the decomposition from a direct numerical simulation are analysed. Finally, the dependence of mixing efficiency on remote regions that was identified by Davies Wykes et al. (J. Fluid Mech., vol. 781, 2015, pp. 261–275) is revisited to demonstrate the precise way in which the proposed decomposition quantifies context.

We consider the interactions of internal tides (ITs) with a dynamic, rotational and heterogeneous ocean, and spatially varying topography. The IT fields are expanded using vertical modal basis functions, whose amplitudes vary horizontally and temporally. We obtain the evolution equations of modal amplitudes and energy including simultaneous three-way interactions with the mean flow, buoyancy and topography. We apply these equations to a set of idealized and two realistic data-assimilative primitive equation simulations. These simulations reveal that significant interactions of ITs with the background fields occur at topographic features and strong currents, in particular when the scales of the background and ITs are similar. In local hot spots, the new three-way interaction terms, when compared to the total modal conversion, are found to reach up to 10 %–30 % at steep topography and approximately 50 % in the Gulf Stream. We provide a dimensional analysis to guide the diagnosis of such strong interactions. When IT interactions are with a large-scale barotropic current (without topographic effects), our modal energy equation reduces to the conservation of modal wave action under a Wentzel–Kramers–Brillouin consideration. We further derive analytical solutions of the modulation of wavenumber and energy of an IT propagating into a collinear current. For ITs propagating along the flow direction, the wavelength is stretched and the amplitude is reduced, with the degree of modulation determined by $|\,f/\omega _0|$, the ratio of inertial to tidal frequencies. For ITs propagating opposite to the flow direction, a critical value of $|\,f/\omega _0|$ exists, below and above which the waves show remarkably different behaviours. The critical opposing current speed which triggers the wave focusing/blocking phenomenon is obtained and its implication for the propagation and dissipation of ITs is discussed.

Large-eddy simulation is utilized to reproduce the instability of the tip vortices shed from the blades of an axial-flow turbine. The oscillations of their helical trajectories trigger mutual interaction between them. This accelerates the process of their destabilization, leading to leapfrogging and eventually to breakdown into smaller structures and loss of coherence, initiating wake contraction and momentum recovery from the outer radii towards the wake core. A strong correlation of the tip vortices instability with the behaviour of the Reynolds stresses and turbulence production is observed. In particular, the turbulent shear stress tied to the fluctuations of the radial and axial velocity components reveals the significant role of the interaction of each tip vortex with the outer region of the wake of the preceding blade, creating a ‘bridge’ between neighbouring tip vortices. Such an interaction enhances the process of mutual inductance between them, promoting production of turbulence and destabilization of the coherent structures. The latter results in increasing oscillations of the radial location of their cores and in a significant jump of the normal turbulent stress of radial velocity within them. Further downstream, the instability of the tip vortices triggers intense mixing phenomena between the outer free stream and the inner wake flow, leading the process of momentum recovery and wake contraction.

We systematically study dissipative anomaly in compressible turbulence using a direct numerical simulations (DNS) database spanning a large parameter space, and show that the classical incompressible scaling does not hold for the total dissipation field. We assess the scaling for the solenoidal and dilatational parts separately. The solenoidal dissipation obeys the same scaling as incompressible turbulence when rescaled on solenoidal variables. We propose new scaling laws for total dissipation that predict the transition between regimes dominated by the solenoidal and dilatational components, and confirm them by the DNS data. An analysis of dilatational dissipation shows that dissipative anomaly may hold if properly scaled for certain regimes; on this empirical basis, we propose a new criterion for the energy cascade in the dilatational component.

The interaction between the flow above and below a permeable wall is a central topic in the study of porous media. While previous investigations have provided compelling evidence of the strong coupling between the two regions, few studies have quantitatively measured the directionality, i.e. cause-and-effect relations, of this interaction. To shed light on the problem, interface-resolved direct numerical simulations of channel flow over a cylinder array for porosity $\varphi =0.4$–$0.9$ are performed, and the friction Reynolds number of the top smooth wall is controlled to be $Re_{\tau }\approx 180$. We use transfer entropy as a marker to evaluate the causal interaction between the free turbulent flow and the porous medium. Correlation analysis and linear coherence spectra are also leveraged to complete the study. Our results show that the permeability of the porous medium has a profound impact on the intensity, time scale and spatial extent of surface–subsurface interactions. The interaction of the free flow and porous medium is further decomposed into two coupling directions, namely, top-down and bottom-up. For low-porosity cases, top-down and bottom-up interactions are strongly asymmetric, the former being mostly influenced by small near-wall eddies, and the latter reflecting the onset of Kelvin–Helmholtz type instabilities due to the perturbation from the porous medium. As the porosity increases, both top-down and bottom-up interactions are dominated by shear-flow instabilities.

In the context of water waves, we consider a resonator with deep subwavelength resonance, analogue to the Helmholtz resonator in acoustics. In the shallow water regime, using asymptotic analysis, a one-dimensional model is derived in which the effect of the resonator is reduced to effective transmission conditions. These conditions clearly highlight two contributions. The first is associated with the dock on its own and it is responsible for a jump of the potential at the free surface. The second is due to the resonant cavity and it is responsible for a jump in the horizontal velocity. It involves as well the uniform amplitude within the resonant cavity with a transient dynamics explicitly given by the equation of a damped oscillator forced by the incident waves. The one-dimensional model is validated in the harmonic regime by comparison to direct two-dimensional numerics. It is shown to reproduce accurately the scattering coefficients and the amplitude within the resonator; interestingly, this remains broadly true for finite water depths. We further inspect the spatio-temporal behaviour of different types of wave packets interacting with the resonating and radiating cavity.

We study the linear stability of a surfactant-laden shear-imposed fluid flowing down a compliant substrate. The aim is to extend the earlier and recent studies (Carpenter & Garrad, J. Fluid Mech., vol. 155, 1985, pp. 465–510; Alexander et al., J. Fluid Mech., vol. 900, 2020, A40) in the presence of insoluble surfactant when an external streamwise imposed shear stress acts at the fluid surface. In other words, the current study expands the earlier study (Wei, Phys. Fluids, vol. 17, 2005, 012103) in the presence of a flexible substrate. The Orr–Sommerfeld-type boundary value problem is derived and solved by using the long-wave series expansion as well as the Chebyshev spectral collocation method for disturbances of arbitrary wavenumbers. The long-wave result reveals the existence of two dominant temporal modes, the so-called surface mode and surfactant mode, where the surface mode propagates faster than the surfactant mode. It is found that the surface mode can be stabilized by introducing an insoluble surfactant at the fluid surface even though the spring stiffness $C_K$ keeps a lower value than its critical value $C_K^*$. But the imposed shear stress exhibits a dual role in the surface mode in two different regimes of spring stiffness $C_K$, i.e. a stabilizing effect when $C_K< C_K^*$ and a destabilizing effect when $C_K>C_K^*$. Further, the surfactant mode becomes more unstable with the increasing values of spring stiffness $C_K$ and damping coefficient $C_D$. On the other hand, the numerical result in the arbitrary wavenumber regime reveals the existence of subcritical instability induced by the surface mode. Furthermore, a different temporal mode, the so-called wall mode, appears in the finite wavenumber regime for special values of $C_K$ and $C_D$, which becomes weaker with increasing values of the wall parameters $C_K$, $C_D$, $C_B$ and $C_T$, but becomes stronger with increasing values of the inclination angle $\theta$ and wall parameter $C_I$. Moreover, the temporal growth rate associated with the wall mode enhances with the increasing value of the Marangoni number but attenuates with the increasing value of imposed shear stress. In addition, another temporal mode, the so-called shear mode, emerges in the finite wavenumber regime when the Reynolds number is high and the inclination angle is small. The unstable region generated by the shear mode magnifies with the increasing value of the imposed shear stress but decays with the increasing value of Marangoni number. Further, the shear mode becomes more unstable as soon as the spring stiffness $C_K$ and damping coefficient $C_D$ increase.

This study investigates the effect of an isothermal wall with complex impedance on compressible turbulent channel flow up to bulk Mach numbers of $6.00$. Such investigation is carried out via the time-domain impedance boundary conditions based on auxiliary differential equations method. A three-parameter complex impedance, modelling a frequency-selective porous wall, with tuneable resonating frequency $\omega _{res}$ and variable resistance $R \in [0.10, 1.0]$ is employed. Higher resistance leads to lower wall permeability with $R \rightarrow \infty$ representing the impermeable limit. Three bulk Mach numbers $M_b = \{1.50, 3.50, 6.00\}$ are investigated with a semi-local Reynolds number $Re_\tau ^{*} \approx 220$. It is found that a sufficiently low $R$ could trigger flow instabilities, which comprise streamwise-travelling waves in the near-wall region, akin to spanwise rollers at low subsonic flow conditions and second-mode waves at hypersonic conditions. The probability density function of instantaneous wall-shear stress shows an enhancement in extreme positive cases of wall-shear stress fluctuations, leading to an increase in the mean wall-shear stress due to porous walls. The wave dynamically affects the turbulence, yielding a local peak near the wall in the pre-multiplied spectrum of the production term of turbulence kinetic energy. Linear stability analysis using the turbulent base flow profile confirmed that the finite wall permeability triggers the instability when $R$ is below a threshold $R_{{cr}}$, which shows a sub-linear proportionality on the bulk Mach number $M_b$. The perturbed field exhibits more dilatational nature in high Mach number flows with low permeability.

The present study deals with the effect of micromagnetorotation (MMR) on a micropolar Poiseuille flow in the presence of a uniform magnetic field. Micromagnetorotation is associated with the impact of magnetization on magnetohydrodynamic (MHD) micropolar flows. Previously, magnetization was assumed to be parallel to the applied magnetic field and thus, its influence on the flow was ignored. This assumption is incorrect in the case of micropolar fluids, because their anisotropy affects magnetization. Here, the velocity and microrotation fields, as well as the skin friction coefficient are examined analytically by using a new MHD micropolar fluid theory that includes a constitutive equation for magnetization. Results reveale that MMR has a strong braking effect both on velocity and microrotation. Flow deceleration is found to be up to 16 %, while an increase in the skin friction coefficient is also observed. Moreover, the stability of the MHD micropolar flow is studied by introducing a modified version of the Orr–Sommerfeld equation, which incorporates MMR. The eigenvalue problem is solved with the use of the open-source Chebfun library. It is found that the MMR has a strong stabilizing effect on the MHD micropolar flow. Thus, the MMR is proved to be a mechanism similar to the Lorentz force, which dissipates additional magnetic energy to the flow via microrotation. In summary, the important effect of MMR, neglected by researchers so far, should be considered for industrial and bioengineering applications that involve micropolar fluids and magnetic fields.

The linear stability of a vertical interface separating two miscible fluid columns of different densities and viscosities under the influence of gravity is investigated. This flow possesses a time-dependent reference state (each column accelerates at different rates owing to their different densities) and the interface thickness grows as the square root of time (by diffusion). Numerical integration of the linear initial-value problem is carried out and discussed in detail as a function of vertical and spanwise wavenumbers and the flow parameters. Adjoint-based optimization is performed in order to determine initial conditions that lead to maximum growth of disturbances in finite time. Results indicate that the rate of growth of the perturbation energy at small wavenumbers (less affected by viscosity initially) is dominated by two-dimensional modes (no spanwise variation). Substantial transient growth is observed at higher wave modes initially, followed by asymptotic decay of the perturbations at large time. Sensitivity of perturbation growth with respect to initial time, density and viscosity ratios is investigated. This work is complementary to previous inviscid analysis of this configuration, which showed that the interface was unconditionally unstable at all wave modes, even in the presence of surface tension, and that instability grew as the exponential of time squared.

Identification of coherent structures is an essential step to describe and model turbulence generation mechanisms in wall-bounded flows. To this end, we present a clustering method based on latent Dirichlet allocation (LDA), a generative probabilistic model for collection of discrete data. The method is applied to a set of snapshots featuring the Reynolds stress ($Q_-$ events) for a turbulent channel flow at a moderate Reynolds number $R_{\tau }=590$. Both two- and three-dimensional analysis show that LDA provides a robust and compact flow description in terms of a combination of motifs, which are latent variables inferred from the set of snapshots. We find that the characteristics of the motifs scale with the wall distance, in agreement with the wall-attached eddy hypothesis of Townsend (Physics of Fluids, 1961, pp. 97–120). Latent Dirichlet allocation motifs can be used to reconstruct fields with an efficiency that can be compared with the proper orthogonal decomposition (POD). Moreover, the LDA model makes it possible to generate a collection of synthetic fields that captures the intermittent characteristics of the original dataset more clearly than its POD-generated counterpart. These findings highlight the potential of LDA for turbulent flow analysis, efficient reconstruction of actual fields and production of a new field with suitable statistics.

Discrete roughness-induced transition in Mach $2.25$ flow controlled by two-dimensional wall blowing is studied using direct numerical simulation. Spectral analysis and flow freezing operations reveal that the main source of unsteadiness in the case without blowing is the separated shear layer/counter-rotating vortex system: the shear layer is bent by the vortex pair, and this interaction induces disturbance growth at the shear layer. With the existence of weak wall blowing, the transition is delayed. Flow visualization demonstrates that upstream-positioned blowing achieves this by lifting the inflow boundary layer and decreasing the roughness Reynolds number. In contrast, downstream-positioned blowing takes effect by weakening the counter-rotating vortex pair and inhibiting the interaction. Vorticity transportation analysis suggests that this result is accomplished by increasing dissipation for streamwise vorticity and converting some of it into spanwise vorticity. In cases with strong wall blowing, the control effect is reversed, as wall blowing with enough strength can induce unsteadiness and promote transition. In upstream-positioned strong blowing cases, a new unstable mode is observed in spectral results throughout the near-roughness region, and further analysis indicates that this mode originating from the separation zone upstream of the blowing is the key factor for transition. With regard to downstream-positioned strong blowing cases, a very low-frequency mode is generated from the separation bubble between the roughness element and the blowing, but it only provides an initial disturbance for the transition process. The transition that occurs downstream is due to the distortion of the boundary layer in the wake.

This paper presents an experimental investigation of the characteristics of laminar– turbulent transition occurring on a flat-plate boundary layer due to the interaction with a non-impinging aerofoil wake. Previous studies have tended to focus on transition induced by free-stream turbulence or by the wake of a circular cylinder, both of which exhibit different forcing characteristics to the present experimental arrangement. A tripped NACA 0014 aerofoil was used to generate a fully turbulent wake, upstream of and at various heights above a laminar, flat-plate boundary layer, in the UK National Low-turbulence Wind Tunnel at City, University of London. Hot-wire measurements conducted in the pre-transitional region reveal the wall-normal and spanwise structure of the disturbances induced within the boundary layer and the rate of growth of disturbance energy. Disturbance profiles generally (but not uniquely) follow the non-modal distribution obtained from transient growth theory, but energy growth rates are mainly exponential rather than algebraic. Energy spectra demonstrate the existence of mixed transitional features (both natural and bypass) in the boundary layer. Two-point spatial correlations reveal the presence of a streaky structure, but with spanwise scale much larger than the boundary layer thickness, in contrast to the trends seen in free-stream turbulence-induced bypass transition and cylinder wake-induced transition. The gap between aerofoil and flat plate affects both the evolution of non-modal disturbance profile and the appearance of the streaky structure; the spacing of the streaks was also found to scale with the vertical gap between aerofoil and flat plate. Overall, the combination of observed characteristics is quite different from the forced transition mechanisms previously reported in the literature.

In this paper, we address the question of whether total helicity is conserved through viscous reconnection events in topologically complex vortex flows. To answer this question, we performed direct numerical simulations (DNS) focused on two complex vortex flow problems: (1) a trefoil knot and (2) a two-ring link, both simulated for various vortex core radii. The DNS framework relies on a block-structured adaptive mesh refinement (AMR) technique. A third simulation of a colliding pair of unlinked vortex rings, which exhibit no total helicity change, is also performed to serve as a reference case. The results show that a well-defined total helicity jump occurs during the unknotting/unlinking events of cases (1) and (2), which arises from the annihilation of the local helicity density content in the reconnection regions. Changes in total helicity become steeper as thinner core radii are considered for both cases (1) and (2). Finally, an analytical derivation based on the reconnection of two infinitesimal anti-parallel vortex filaments is provided that quantitatively links helicity annihilation and viscous circulation transfer processes, which unveils the fundamental hydrodynamic mechanisms responsible for production/destruction of total helicity during reconnection events.

The scale-space energy density function $E(\boldsymbol {x},\boldsymbol {r})$ is defined as the derivative of the two-point velocity correlation $Q_{ii}(\boldsymbol {x},\boldsymbol {r})$ as $E(\boldsymbol {x},r_\alpha ) = -(\partial Q_{ii}(\boldsymbol {x},\boldsymbol {r})/\partial r_\alpha )/{2}$, where $\boldsymbol {x}$ is the spatial coordinate of interest and $\boldsymbol {r}$ is the separation vector. The function $E$ describes the turbulent kinetic energy density of scale $|\boldsymbol {r}|$ at a location $\boldsymbol {x}$ and can be considered as the generalization of the spectral energy density function concept to inhomogeneous flows. In this work, we derive the scale-space energy density function transport equation for compressible flows to develop a better understanding of scale-to-scale energy transfer and the degree of non-locality of the energy interactions. Specifically, the effects of variable-density and dilatation on an energy cascade are identified. It is expected that these findings will yield deeper insight into compressibility effects on canonical energy cascades, which will lead to improved models (at all levels of closure) for mass flux, density variance, pressure-dilatation, pressure–strain correlation and dilatational dissipation processes. Direct numerical simulation (DNS) data of mixing layers at different Mach numbers are used to characterize the scale-space behaviour of different turbulence processes. The scaling of the energy density function that leads to self-similar evolution at the two Mach numbers is identified. The scale-space (non-local) behaviour of the production and pressure dilatation at the centre-plane is investigated. It is established that production is influenced by long-distance (order of vorticity thickness) interactions, whereas the pressure dilatation effects are more localized (fraction of momentum thickness) in scale space. The analysis of DNS data demonstrates the utility of the energy density function and its transport equation to account for the relevance of various physical mechanisms at different scales.

Oceanic mesoscale currents (‘eddies’) can have significant effects on the distributions of passive tracers. The associated inhomogeneous and anisotropic eddy fluxes are traditionally parametrised using a transport tensor ($K$-tensor), which contains both diffusive and advective components. In this study, we analyse the eddy transport tensor in a quasigeostrophic double-gyre flow. First, the flow and passive tracer fields are decomposed into large- and small-scale (eddy) components by spatial filtering, and the resulting eddy forcing includes an eddy tracer flux representing advection by eddies and non-advective terms. Second, we use the flux-gradient relation between the eddy fluxes and the large-scale tracer gradient to estimate the associated $K$-tensors in their entire structural, spatial and temporal complexity, without making any additional assumptions or simplifications. The divergent components of the eddy tracer fluxes are extracted via the Helmholtz decomposition, which yields a divergent tensor. The remaining rotational flux does not affect the tracer evolution, but dominates the total tracer flux, affecting both its magnitude and spatial structure. However, in terms of estimating the eddy forcing, the transport tensor prevails over its divergent counterpart because of the significant numerical errors induced by the Helmholtz decomposition. Our analyses demonstrate that, in general, the $K$-tensor for the eddy forcing is not unique, that is, it is tracer-dependent. Our study raises serious questions on how to interpret and use various estimates of $K$-tensors obtained from either observations or eddy-resolving model solutions.

The momentum and energy exchanges at the ocean surface are central factors determining the sea state, weather patterns and climate. To investigate the effects of surface waves on the air–sea energy exchanges, we analyse high-resolution laboratory measurements of the airflow velocity acquired above wind-generated surface waves using the particle image velocimetry technique. The velocity fields were further decomposed into the mean, wave-coherent and turbulent components, and the corresponding energy budgets were explored in detail. We specifically focused on the terms of the budget equations that represent turbulence production, wave production and wave–turbulence interactions. Over wind waves, the turbulent kinetic energy (TKE) production is positive at all heights with a sharp peak near the interface, indicating the transfer of energy from the mean shear to the turbulence. Away from the surface, however, the TKE production approaches zero. Similarly, the wave kinetic energy (WKE) production is positive in the lower portion of the wave boundary layer (WBL), representing the transfer of energy from the mean flow to the wave-coherent field. In the upper part of the WBL, WKE production becomes slightly negative, wherein the energy is transferred from the wave perturbation to the mean flow. The viscous and Stokes sublayer heights emerge as natural vertical scales for the TKE and WKE production terms, respectively. The interactions between the wave and turbulence perturbations show an energy transfer from the wave to the turbulence in the bulk of the WBL and from the turbulence to the wave in a thin layer near the interface.

Bubble-mediated gas exchange in turbulent flow is critical in bubble column chemical reactors as well as for ocean–atmosphere gas exchange related to air entrained by breaking waves. Understanding the transfer rate from a single bubble in turbulence at large Péclet numbers (defined as the ratio between the rate of advection and diffusion of gas) is important as it can be used for improving models on a larger scale. We characterize the mass transfer of dilute gases from a single bubble in a homogeneous isotropic turbulent flow in the limit of negligible bubble volume variations. We show that the mass transfer occurs within a thin diffusive boundary layer at the bubble–liquid interface, whose thickness decreases with an increase in turbulent Péclet number, $\widetilde {{Pe}}$. We propose a suitable time scale $\theta$ for Higbie (Trans. AIChE, vol. 31, 1935, pp. 365–389) penetration theory, $\theta = d_0/\tilde {u}$, based on $d_0$ the bubble diameter and $\tilde {u}$ a characteristic turbulent velocity, here $\tilde {u}=\sqrt {3}\,u_{{rms}}$, where $u_{{rms}}$ is the large-scale turbulence fluctuations. This leads to a non-dimensional transfer rate ${Sh} = 2(3)^{1/4}\sqrt {\widetilde {{Pe}}/{\rm \pi} }$ from the bubble in the isotropic turbulent flow. The theoretical prediction is verified by direct numerical simulations of mass transfer of dilute gas from a bubble in homogeneous and isotropic turbulence, and very good agreement is observed as long as the thin boundary layer is properly resolved.

We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh–Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy–Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection in both the porous and the unconfined fluid regions, while the porous flow still remains strongly confined and governed by Darcy's law. We demonstrate that the heat flux and average thermal structure of the system can be predicted using previous results of convection in individual fluid or porous layers. We revisit a controversy about the role of subcritical ‘penetrative convection’ in the porous medium, and confirm that such induced flow does not contribute to the heat flux through the system. Lastly, we briefly study the temporal coupling between the two layers and find that the turbulent fluid convection above acts as a low-pass filter on the longer time-scale variability of convection in the porous layer.

This paper describes the development of a method of curved-shock characteristics based on curved shock theory. The proposed method is applied to supersonic flowfield calculations and inverse design in planar/axisymmetric, external/internal and uniform/non-uniform flows. The main idea is to determine the gradients of the pressure and flow deflection angle in the streamline-characteristic coordinates. With the acquired derivatives, the flow parameters of the post-shock flowfield can be quickly identified. Compared with the method of characteristics, the gradient information enhances the computational efficiency and accuracy of the method of curved-shock characteristics. This makes the method of curved-shock characteristics more effective, accurate and robust than the conventional method of characteristics. Explicit equations in the form of gradients are derived along the streamlines and characteristics. Several supersonic flowfields are solved using the method of curved-shock characteristics, and the results show that the proposed method requires less computational resources, by an order of magnitude, than the method of characteristics while achieving superior accuracy. Additionally, the proposed method is applied to the inverse design of internal flows. A series of planar and axisymmetric flowfields with centre-bodies are solved under the condition that the shock curves are given. The accuracy and efficiency of the method of curved-shock characteristics make it a good candidate for the inverse design of planar/axisymmetric supersonic flowfields.