Skip to main content Accessibility help
×
  • Cited by 14
  • Kun Xu, Hong Kong University of Science and Technology
Publisher:
Cambridge University Press
Online publication date:
May 2021
Print publication year:
2021
Online ISBN:
9781108877534

Book description

This Element presents a unified computational fluid dynamics framework from rarefied to continuum regimes. The framework is based on the direct modelling of flow physics in a discretized space. The mesh size and time step are used as modelling scales in the construction of discretized governing equations. With the variation-of-cell Knudsen number, continuous modelling equations in different regimes have been obtained, and the Boltzmann and Navier-Stokes equations become two limiting equations in the kinetic and hydrodynamic scales. The unified algorithms include the discrete velocity method (DVM)–based unified gas-kinetic scheme (UGKS), the particlebased unified gas-kinetic particle method (UGKP), and the wave and particle–based unified gas-kinetic wave-particle method (UGKWP). The UGKWP is a multi-scale method with the particle for non-equilibrium transport and wave for equilibrium evolution. The particle dynamics in the rarefied regime and the hydrodynamic flow solver in the continuum regime have been unified according to the cell's Knudsen number.

References

Alexander, F.J., Garcia, A.L., & Alder, B.J. (1998). Cell size dependence of transport coefficients in stochastic particle algorithm. Phys. Fluids, 10, 15401542.
Aristov, V. (2012). Direct methods for solving the Boltzmann equation and study of nonequilibrium flows. Springer Science & Business Media.
Bhatnagar, P.L., Gross, E.P., & Krook, M. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511.
Bird, G. (1970). Aspects of the structure of strong shock waves. Phys. Fluids 13, 11721177.
Bird, G. (1994). Molecular gas dynamics and the direct simulation of gas flows. Oxford Science Publications.
Burt, J., Josyula, E., Deschenes, T., & Boyd, I. (2011). Evaluation of a hybrid Boltzmann-continuum method for high-speed nonequilibrium flows. J. Thermophys. Heat Trans. 25, 500515.
Chacon, L., Chen, G., Knoll, D., Newman, C., Park, H., Taitano, W., Willert, J., & Womeldorff, G. (2017). Multiscale high-order/low-order (HOLO) algorithms and applications. J. Comput. Phys. 330, 2145.
Chapman, S., Cowling, T.G., & Burnett, D. (1990). The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press.
Chen, S., & Doolen, G. (1998). Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. 30, 329364.
Chen, S.Z., Xu, K., & Cai, Q. (2015). A comparison and unification of ellipsoidal statistical and Shakhov BGK. Adv. Appl. Math. Mech. 7, 245266.
Chen, S.Z., Xu, K., Lee, C.B., & Cai, Q. (2012). A unified gas kinetic scheme with moving mesh and velocity space adaptation. J. Comput. Phys. 231 (20), 66436664.
Chen, S.Z, Zhang, C., Zhu, L., & Guo, Z. (2017). A unified implicit scheme for kinetic model equations. part I. memory reduction technique. Science Bull. 62 (2), 119129.
Chen, Y.P., Zhu, Y.J., & Xu, K. (2020). A three-dimensional unified gas-kinetic wave-particle solver for flow computation in all regimes. Phys. Fluids 32, 096108.
Chou, S., & Baganoff, D. (1997). Kinetic flux-vector splitting for the Navier-Stokes equations. J. Comput. Phys. 130, 217230.
Chu, C.K. (1965). Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8 (1), 1222.
Degond, P., Dimarco, G., & Mieussens, L. (2010). A multiscale kinetic–fluid solver with dynamic localization of kinetic effects. J. Comput. Phys. 229 (13), 49074933.
Degond, P., Liu, J., & Mieussens, L. (2006). Macroscopic fluid models with localized kinetic upscale effects. Multiscale Model. Simul. 5, 940979.
Deshpande, S. (1986). A second order accurate, kinetic-theory based method for inviscid compressible flows. NASA Langley Tech. Paper No. 2613.
Dimarco, G., & Pareschi, L. (2013). Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal. 51, 10641087.
Eu, B. (2016). Kinetic theory of nonequilibrium ensembles, irreversible thermodynamics, and generalized hydrodynamics: Volume 1. Nonrelativistic theories. Springer.
Fei, F., Zhang, J., Li, J., & Liu, Z. (2020). A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows. J. Comput. Phys. 400, 108972.
Filbet, F., & Jin, S. (2010). A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229 (20), 76257648.
Gallis, M., & Torczynski, J. (2000). The application of the BGK model in particle simulations. In 34th Thermophysics Conference, 2360.
Ghia, U., Ghia, K., & Shin, C. (1982). High-Resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387411.
Gorji, M.H., & Jenny, P. (2015). Fokker-Planck-DSMC algorithm for simulations of rarefied gas flows. J. Comput. Phys. 287, 110129.
Grad, H. (1949). On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 325.
Gu, X., & Emerson, D. (2009). A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177216.
Guo, Z., Li, J., & Xu, K. (2020). On unified preserving properties of kinetic schemes. arXiv: 1909.04923v4.
Guo, Z., Wang, R., & Xu, K. (2015). Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case. Phys. Rev. E 91 (3), 033313.
Guo, Z., Xu, K., & Wang, R. (2013). Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case. Phys. Rev. E 88 (3), 033305.
Huang, J.C., Xu, K., & Yu, P. (2012). A unified gas-kinetic scheme for continuum and rarefied flows II: Multidimensional cases. Comm. Comput. Phys. 12 (3), 662690.
Huang, J.C., Xu, K., & Yu, P. (2013). A unified gas-kinetic scheme for continuum and rarefied flows III: Microflow simulations. Comm. Comput. Phys. 14 (5), 11471173.
Jenny, P., Torrilhon, M., & Heinz, S. (2010). A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion. J. Comput. Phys. 229 (4) 10771098.
Ji, X., Pan, L., Shyy, W., & Xu, K. (2018b). A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 372, 446472.
Ji, X., Zhao, F., Shyy, W., & Xu, K. (2018a). A family of high-order gas-kinetic schemes and its comparison with Riemann solver based high-order methods. J. Comput. Phys. 356, 150173.
Ji, X., Zhao, F., Shyy, W., & Xu, K. (2020). A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured meshes. J. Comput. Phys. 410, 109367.
Jiang, D.W., Mao, M.L., & Deng, X.G. (2019a). An implicit parallel UGKS solver for flows covering various regimes. Adv. Aerodynamics 1:8, https://doi.org/10.1186/s42774-019-0008-5.
Jiang, Z., Zhao, W., Yuan, Z., Chen, W., & Myong, R. (2019b). Computation of hypersonic flows over flying configurations using a nonlinear constitutive model. AIAA J. 57 (12), 52525268.
Jin, C., & Xu, K. (2007). A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation. J. Comput. Phys. 222, 155175.
Jin, S. (1999). Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comp. 21 (2), 441454.
John, B., Gu, X., & Emerson, D. (2011). Effects of incomplete surface accommodation on non-equilibrium heat transfer in cavity flow: A parallel DSMC study. Comput. Fluids 45, 197201.
Kolobov, V., Arslanbekov, R., Aristov, V., Frolova, A., & Zabelok, S. (2007). Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J. Comput. Phys. 223, 589608.
Kumar, G., Girimaji, S., & Kerimo, J. (2013). WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence. J. Comput. Phys. 234, 499523.
Larsen, A., Morel, J., & Miller, W. (1987). Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys. 69(2),283324.
Lele, S. (1992). Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 1642.
Levermore, C. (1996). Moment closure hierarchies for kinetic theories. J. Statistical Phys. 83, 1021.
Li, J., & Du, Z. (2016). A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers I: Hyperbolic conservation laws. SIAM J. Sci. Comput. 38 (5), A3046A3069.
Li, Q., Xu, K., & Fu, S. (2010). A high-order gas-kinetic Navier-Stokes solver. J. Comput. Phys. 229, 67156731.
Li, S., Li, Q., Fu, S., & Xu, K. (2018). A unified gas-kinetic scheme for axisymmetric flow in all Knudsen number regimes. J. Comput. Phys. 366, 144169.
Li, W., Liu, C., Zhu, Y., Zhang, J., & Xu, K. (2020). Unified gas-kinetic wave-particle methods III: Multiscale photon transport. J. Comput. Phys. 408, 109280.
Li, Z.H., & Zhang, H.X. (2004). Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. J. Comput. Phys. 193 (2), 708738.
Liu, C. (2016). Unified gas-kinetic scheme for the study of multi-scale flows. PhD thesis, Hong Kong University of Science and Technology.
Liu, C., Wang, Z., & Xu, K. (2019a). A unified gas-kinetic scheme for continuum and rarefied flows VI: Dilute disperse gas-particle multiphase system. J. Comput. Phys. 386, 264295.
Liu, C., & Xu, K. (2017). A unified gas kinetic scheme for continuum and rarefied flows V: Multiscale and multicomponent plasma transport. Comm. Comput. Phys. 22 (5), 11751223.
Liu, C., & Xu, K. (2020a). A unified gas-kinetic scheme for micro flow simulation based on linearized kinetic equation. Adv. Aerodynamics 2:21, https://doi.org/10.1186/s42774-020-00045-8.
Liu, C., & Xu, K. (2021). Unified gas-kinetic wave-particle methods IV: Multi-species gas mixture and plasma transport. Adv. Aerodynamics 3:9, https://doi.org/10.1186/s42774-021-00062-1
Liu, C., Xu, K., Sun, Q., & Cai, Q. (2016). A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations. J. Comput. Phys. 314, 305–40.
Liu, C., Zhou, G., Shyy, W., & Xu, K. (2019b). Limitation principle for computational fluid dynamics. Shock Waves 29:10831102.
Liu, C., Zhu, Y., & Xu, K. (2020). Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow. J. Comput. Phys. 401, 108977.
Liu, S., Yu, P., Xu, K., & Zhong, C. (2014). Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes. J. Comput. Phys. 259, 96113.
Luo, J., & Xu, K. (2013). A high-order multidimensional gas-kinetic scheme for hydrodynamic equations. Science China, Technological Sciences 56, 23702384.
Luo, J., Xuan, L., & Xu, K. (2013). Comparison of fifth-order WENO scheme and WENO-gas-kinetic scheme for inviscid and viscous flow simulation. Commun. Comput. Phys. 14, 599620.
Macrossan, M.N. (2001). A particle simulation method for the BGK equation. In AIP Conference Proceedings, Vol. 585, AIP, 426433.
Mieussens, L. (2000). Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Applied Sci. 10 (8), 11211149.
Mieussens, L. (2013). On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic model. J. Comput. Phys. 253, 138156.
Mouhot, C., & Pareschi, L. (2006). Fast algorithms for computing the Boltzmann collision operator. Math. Comp. 75 (256), 18331852.
Myong, R. (2001). A computational method for Eu’s generalized hydrodynamic equations of rarefied and microscale gas dynamics. J. Comput. Phys. 168, 4772.
Ohwada, T., Adachi, R., Xu, K., & Luo, J. (2013). On the remedy against shock anomalies in kinetic schemes. J. Comput. Phys. 255, 106129.
Ohwada, T., & Kobayashi, S. (2004). Management of discontinuous reconstruction in kinetic schemes. J. Comput. Phys. 197, 116138.
Oran, E., Oh, C., & Cybyk, B. (1998). Direct simulation Monte Carlo: Recent advances and applications. Ann. Rev. Fluid Mech. 30 (1), 403441.
Pan, L., & Xu, K. (2016). A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier–Stokes solutions. J. Comput. Phys. 318, 327348.
Pan, L., Xu, K., Li, Q., & Li, J. (2016). An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 326, 197221.
Pareschi, L., & Russo, G. (2000). Asymptotic preserving Monte Carlo methods for the Boltzmann equation. Trans. Theory Stat. Phys. 29 (3–5) 415430.
Perthame, B. (1992). Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29, 119.
Pieraccini, S., & Puppo, G. (2007). Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput. 32, 128.
Pullin, D. (1980). Direct simulation methods for compressible inviscid ideal gas flow. J. Comput. Phys. 34, 231244.
Ren, W., Liu, H., & Jin, S. (2014). An asymptotic-preserving Monte Carlo method for the Boltzmann equation. J. Comput. Phys. 276, 380404.
Ren, X.D., Xu, K., & Shyy, W. (2016). A multidimensional high-order DG-ALE method based on gas-kinetic theory with application to oscillating bodies. J. Comput. Phys. 316, 700720.
Schwartzentruber, T.E., & Boyd, I. (2007). A molecular particle-continuum method for hypersonic nonequilibrium gas flows. J. Comput. Phys. 225, 11591174.
Schwartzentruber, T.E., Scalabrin, L.C., & Boyd, I. (2008). Hybrid particle-continuum simulations of nonequilibrium hypersonic blunt-body flowfields. J Thermophys. Heat Trans. 22 (1), 2937.
Shakhov, E. (1968). Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3 (5), 9596.
Shi, Y., Song, P., & Sun, W. (2020). An asymptotic preserving unified gas kinetic particle method for radiative transfer equations. J. Comput. Phys. 420, 109687.
Shi, Y., Sun, W., Li, L., & Song, P. (2021). An improved unified gas kinetic particle method for radiative transfer equations. J. Quant. Spectrosc. Radiat. Transfer 261,107428.
Shizgal, B. (1981). A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. J. Comput. Phys. 41, 309328.
Struchtrup, H. (2005). Macroscopic transport equations for rarefied gas flows: Approximation methods in kinetic theory. Springer.
Struchtrup, H., & Torrihon, M. (2003). Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids 15, 2668.
Su, W., Zhu, L., Wang, P., Zhang, Y., & Wu, L. (2020). Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations? J. Comput. Phys. 407, 109245.
Succi, S. (2015). Lattice Boltzmann 2038. EPL 109 (5), 50001.
Sun, W., Jiang, S., & Xu, K. (2015a). An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations. J. Comput. Phys. 285, 265279.
Sun, W., Jiang, S., & Xu, K. (2017). A multidimensional unified gas-kinetic scheme for radiative transfer equations on unstructured mesh. J. Comput. Phys. 351, 455472.
Sun, W., Jiang, S., Xu, K., & Li, S. (2015b). An asymptotic preserving unified gas kinetic scheme for frequency dependent radiative transfer equations. J. Comput. Phys. 302, 222238.
Tan, S., Sun, W., Wei, J., & Ni, G. (2019). A parallel unified gas kinetic scheme for three-dimensional multi-group neutron transport. J. Comput. Phys. 391, 3758.
Tcheremissine, F. (2005). Direct numerical solution of the Boltzmann equation. In AIP Conference Proceedings, Vol. 762, AIP, 677685.
Tiwari, S. (1998). Coupling of the Boltzmann and Euler equations with automatic domain decomposition. J. Comput. Phys. 144, 710726.
Tumuklu, O., Li, Z., & Levin, D.A. (2016). Particle ellipsoidal statistical Bhatnagar-Gross-Krook approach for simulation of hypersonic shocks. AIAA J. 3701–3716.
van Leer, B. (1977). Towards the ultimate conservative difference scheme IV: a new approach to numerical convection. J. Comput. Phys. 23, 276299.
van Leer, B. (1979). Towards the ultimate conservative difference scheme V: A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101136.
Vincenti, W., & Kruger, C. (1965). Introduction to physical gas dynamics. Krieger.
Wagner, W. (1992). A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation. J. Statistical Phys. 66, 10111044.
Wang, Z., Yan, H., Li, Q., & Xu, K. (2017). Unified gas-kinetic scheme for diatomic molecular flow with translational, rotational, and vibrational modes. J. Comput. Phys. 350, 237259.
Wijesinghe, H.S., & Hadjiconstantinou, N. (2004). Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics. J. Fluid Engineering 126, 768777.
Woodward, P., & Colella, P. (1984). Numerical simulations of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115173.
Wu, L., Reese, J.M., & Zhang, Y. (2014). Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows. J. Fluid Mech. 746, 5384.
Wu, L., White, C., Scanlon, T.J., Reese, J.M., & Zhang, Y. (2013). Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J. Comput. Phys. 250, 2752.
Wu, L., Zhang, J., Reese, J.M., & Zhang, Y. (2015). A fast spectral method for the Boltzmann equation for monatomic gas mixtures. J. Comput. Phys. 298, 602621.
Xiao, T., Cai, Q., & Xu, K. (2017). A well-balanced unified gas-kinetic scheme for multiscale flow transport under gravitational field. J. Comput. Phys. 332, 475491.
Xiao, T., Liu, C., Xu, K., & Cai, Q. (2020). A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows. J. Comput. Phys. 415, 109535.
Xu, K. (2001). A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171 (1), 289335.
Xu, K. (2015). Direct modelling for computational fluid dynamics: Construction and application of unified gas-kinetic scheme. World Scientific.
Xu, K., He, X., & Cai, C. (2008). Multiple temperature kinetic model and gas-kinetic method for hypersonic nonequilibrium flow computations. J. Comput. Phys. 227, 67796794.
Xu, K., & Huang, J.C. (2010). A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys. 229 (20), 77477764.
Xu, K., & Huang, J.C. (2011). An improved unified gas-kinetic scheme and the study of shock structures. IMA J. App. Math. 76, 698711.
Xu, K., & Li, Z. (2001). Dissipative mechanism in Godunov-type schemes. Int. J. Numer. Meth. Fluids 37, 122.
Xu, K., & Liu, C. (2017). A paradigm for modelling and computation of gas dynamics. Phys. Fluids 29, 026101.
Xu, K., Mao, M., & Tang, L. (2005). A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow. J. Comput. Phys. 203, 405421.
Xu, X., Chen, Y., Liu, C., Li, Z., & Xu, K. (2020). Unified gas-kinetic wave-particle methods V: Diatomic molecular flow. arXiv:2010.07195v1 [physics.comp-ph] 14 Oct.
Xu, X., Chen, Y., & Xu, K. (2021). Modelling and computation of non-equilibrium gas dynamics: Beyond single relaxation time kinetic models. Phys. Fluids 33, 011703; doi: 10.1063/5.0036203.
Yang, J., & Huang, J. (1995). Rarefied flow computations using nonlinear model Boltzmann equations. J. Comput. Phys. 120 (2), 323339.
Yang, L., Shu, C., Yang, W., & Wu, J. (2018). An implicit scheme with memory reduction technique for steady state solutions of DVBE in all flow regimes. Phys. Fluids 30 (4), 040901.
Yu, P. (2013). A unified gas kinetic scheme for all Knudsen number flows. PhD thesis, Hong Kong University of Science and Technology.
Zhao, F., Ji, X., Shyy, W., & Xu, K. (2019). Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations. Adv. Aerodynamics 1:13, https://doi.org/10.1186/s42774-019-0015-6.
Zhao, F., Ji, X., Shyy, W., & Xu, K. (2020a). An acoustic and shock wave capturing compact high-order gas-kinetic scheme with spectral-like resolution. Int. J. Comput. Fluid. Dyn. https://doi.org/10.1080/10618562.2020.1821879.
Zhao, F., Ji, X., Shyy, W., & Xu, K. (2020b). Compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations. arXiv:2010.05717v2 [math.NA] 19 Oct 2020.
Zhong, X.L., MacCormack, R.W., & Chapman, D. (1993). Stabilization of the Burnett equations and application to hypersonic flows. AIAA J. 31 (1), 1036.
Zhu, L., Guo, Z., & Xu, K. (2016). Discrete unified gas kinetic scheme on unstructured meshes. Comp. Fluids 127, 211225.
Zhu, L., Pi, X., Su, W., Li, Z.H., Zhang, Y., & Wu, L. (2021). General synthetic iteration scheme for non-linear gas kinetic simulation of multi-scale rarefied gas flows. J. Comput. Phys. 430, 110091.
Zhu, Y.J., Liu, C., Zhong, C.W., & Xu, K. (2019b). Unified gas-kinetic wave-particle methods II: Multiscale simulation on unstructured mesh. Phys. Fluids 31, 067105.
Zhu, Y.J., Zhong, C., & Xu, K. (2017a). Implicit unified gas-kinetic scheme for steady state solution in all flow regimes. J. Comput. Phys. 315, 1638.
Zhu, Y.J., Zhong, C., & Xu, K. (2017b). Unified gas-kinetic scheme with multigrid convergence for rarefied flow study. Phys. Fluids 29, 096102.
Zhu, Y.J., Zhong, C., & Xu, K. (2019a). An implicit unified gas-kinetic scheme for unsteady flow in all Knudsen regimes. J. Comput. Phys. 386, 190217.
Zhu, Y.J., Zhong, C.W., & Xu, K. (2020). Ray effect in rarefied flow simulation. J. Comput. Phys. 422, 109751.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.

Accessibility standard: Unknown

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.