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Determination of Numerical Errors in Under-Resolved DNS of Turbulent Non-isothermal Flows
Flow, Turbulence and Combustion ( IF 2.0 ) Pub Date : 2020-07-09 , DOI: 10.1007/s10494-020-00190-6
S. Yigit , J. Hasslberger , M. Klein

The methodology to quantify the numerical dissipation in Under-resolved DNS (UDNS) based on the balance of the kinetic energy equation by Schranner et al. (Comput Fluids 114:84–97, 2015), has been examined in this study. Furthermore, this methodology has been extended for considering active scalars, based on both balance of the kinetic energy and thermal variance equations for assessing the quality of UDNS. As a first step towards the analysis of reactive flows, where temperature is actively coupled to the momentum equation, the turbulent Rayleigh–Bénard convection in a cubical enclosure is considered here. The simulations have been carried out for different Prandtl numbers ( $$Pr=0.1, 1.0, 10$$ P r = 0.1 , 1.0 , 10 ) to analyse different levels of momentum-temperature coupling for a representative value of nominal Rayleigh number ( $$Ra=10^7$$ R a = 10 7 ) in order to investigate the effects of grid resolution and discretisation schemes on the numerical dissipation. It has been found that the numerical dissipation for both kinetic energy ( $$R_{k}$$ R k ) and thermal variance ( $$R_{t}$$ R t ) can range between positive or negative values depending on the combined effects of temporal and spatial discretisation schemes and grid spacing. The positive values of $$R_{k}$$ R k and $$R_{t}$$ R t indicate a loss of the kinetic energy and thermal variance due to numerical errors, whereas the negative values of $$R_{k}$$ R k and $$R_{t}$$ R t result in a moderate attenuation of these quantities. Accordingly, the numerical dissipation results obtained here (i.e. $$R_{k}$$ R k and $$R_{t}$$ R t ) have been utilised for evaluating the effective values of the Rayleigh and Prandtl number (i.e. $$Ra_{eff}$$ R a eff , $$Pr_{eff}$$ P r eff ) for UDNS. Using the scaling of the mean Nusselt number ( $$Nu \sim {Ra^a} {Pr^b}$$ N u ∼ R a a P r b ), it has been shown that the error relative to exact mean Nusselt number can be estimated based on $$R_{k}$$ R k , $$R_{t}$$ R t and the effective mean Nusselt number of the UDNS ( $$Nu_{UDNS}$$ N u UDNS ). Overall, this methodology has been found a promising tool for the quality assessment of UDNS for the applications of non-isothermal flows (e.g. Rayleigh–Bénard convection).

中文翻译:

确定湍流非等温流的欠解析 DNS 中的数值误差

Schranner 等人基于动能方程的平衡对欠解析 DNS (UDNS) 中的数值耗散进行量化的方法。(Comput Fluids 114:84–97, 2015),已在本研究中得到检验。此外,这种方法已经扩展到考虑活动标量,基于动能平衡和热方差方程来评估 UDNS 的质量。作为反应流分析的第一步,其中温度主动耦合到动量方程,这里考虑立方体外壳中的湍流瑞利-贝纳对流。对不同的 Prandtl 数进行了模拟( $$Pr=0.1, 1.0, 10$$ P r = 0.1 , 1.0 , 10 ) 分析标称瑞利数代表值 ($$Ra=10^7$$R a = 10 7 ) 的不同动量-温度耦合水平,以研究网格分辨率和离散化方案对数值的影响耗散。已经发现,动能 ( $$R_{k}$$ R k ) 和热方差 ( $$R_{t}$$ R t ) 的数值耗散可以介于正值或负值之间,具体取决于组合时间和空间离散化方案和网格间距的影响。$$R_{k}$$ R k 和$$R_{t}$$ R t 的正值表示由于数值误差引起的动能和热方差的损失,而$$R_{k 的负值表示}$$ R k 和$$R_{t}$$ R t 导致这些量的适度衰减。因此,此处获得的数值耗散结果(即 $$R_{k}$$ R k 和 $$R_{t}$$ R t ) 已用于评估瑞利数和普朗特数的有效值(即 $$Ra_{eff}$$R a eff , $$Pr_{eff}$$ P r eff ) 用于 UDNS。使用平均努塞尔特数的标度 ( $$Nu \sim {Ra^a} {Pr^b}$$ N u ∼ R aa P rb ),已经表明相对于精确平均努塞尔特数的误差可以是根据 $$R_{k}$$ R k 、$$R_{t}$$ R t 和 UDNS 的有效平均努塞尔数 ( $$Nu_{UDNS}$$ N u UDNS ) 估计。总体而言,这种方法已被发现是一种很有前途的工具,可用于对非等温流(例如瑞利-贝纳德对流)的应用进行 UDNS 质量评估。使用平均努塞尔特数的标度 ( $$Nu \sim {Ra^a} {Pr^b}$$ N u ∼ R aa P rb ),已经表明相对于精确平均努塞尔特数的误差可以是根据 $$R_{k}$$ R k 、$$R_{t}$$ R t 和 UDNS 的有效平均努塞尔数 ( $$Nu_{UDNS}$$ N u UDNS ) 估计。总体而言,这种方法已被发现是一种很有前途的工具,可用于对非等温流(例如瑞利-贝纳德对流)的应用进行 UDNS 质量评估。使用平均努塞尔特数的标度 ( $$Nu \sim {Ra^a} {Pr^b}$$ N u ∼ R aa P rb ),已经表明相对于精确平均努塞尔特数的误差可以是根据 $$R_{k}$$ R k 、$$R_{t}$$ R t 和 UDNS 的有效平均努塞尔数 ( $$Nu_{UDNS}$$ N u UDNS ) 估计。总体而言,这种方法已被发现是一种很有前途的工具,可用于对非等温流(例如瑞利-贝纳德对流)的应用进行 UDNS 质量评估。
更新日期:2020-07-09
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