Weft knitting is a versatile and widely used machine knitting technology. However, research on computer algorithms for analysis and design of knitted fabrics is limited. This paper addresses important challenges surrounding the design of weft-knitted fabrics with specific flow characteristics. We propose algorithms to generate a yarn topology graph, simulate flow diffusion over the graph, and with a stochastic optimization method generate stitch patterns that produce knitted fabrics with desired flow properties. Our computational results demonstrate the feasibility of using yarn topology graphs to simulate and design flow properties in weft-knitted fabrics.